[ { "id": "Algebra_0000", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Properties", "level": "2", "keywords": [ "commutative" ], "problem_v1": "Use the commutative property of addition to write an equivalent expression.\n${m+59}$ [ANS]", "answer_v1": [ "59+m" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Use the commutative property of addition to write an equivalent expression.\n${y+94}$ [ANS]", "answer_v2": [ "94+y" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Use the commutative property of addition to write an equivalent expression.\n${a+61}$ [ANS]", "answer_v3": [ "61+a" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0001", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Properties", "level": "1", "keywords": [ "distributive", "expression", "linear", "fraction" ], "problem_v1": "Use the distributive property to simplify the expression completely.\n${{\\textstyle\\frac{7}{8}}\\!\\left(-4+{\\textstyle\\frac{4}{7}}m\\right)}$ $=$ [ANS]", "answer_v1": [ "1/2*m-7/2" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Use the distributive property to simplify the expression completely.\n${{\\textstyle\\frac{10}{3}}\\!\\left(-3+{\\textstyle\\frac{5}{2}}y\\right)}$ $=$ [ANS]", "answer_v2": [ "25/3*y-10" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Use the distributive property to simplify the expression completely.\n${{\\textstyle\\frac{7}{4}}\\!\\left(1+{\\textstyle\\frac{3}{5}}a\\right)}$ $=$ [ANS]", "answer_v3": [ "21/20*a+7/4" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0002", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Properties", "level": "3", "keywords": [ "formula", "expression", "evaluate" ], "problem_v1": "Choose True or False for the following questions about the difference between expressions and equations.\n$\\text{We can evaluate}5x+2=2x+5\\text{when}x=1$ [ANS]\n$\\text{We can evaluate}5x+2\\text{when}x=1$ [ANS]\n$5x+2=2x+5\\text{is an expression.}$ [ANS]\n$2x+5\\text{is an equation.}$ [ANS]\n$5x+2=2x+5\\text{is an equation.}$ [ANS]\n$\\text{We can check whether}x=1\\text{is a solution of}5x+2=2x+5.$ [ANS]\n$\\text{We can check whether}x=1\\text{is a solution of}5x+2.$ [ANS]\n$5x+2\\text{is an expression.}$ [ANS]", "answer_v1": [ "False", "True", "False", "False", "True", "True", "False", "True" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v1": [ [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ] ], "problem_v2": "Choose True or False for the following questions about the difference between expressions and equations.\n$-9x+9=9x-9\\text{is an equation.}$ [ANS]\n$-9x+9=9x-9\\text{is an expression.}$ [ANS]\n$\\text{We can check whether}x=1\\text{is a solution of}-9x+9.$ [ANS]\n$\\text{We can evaluate}-9x+9=9x-9\\text{when}x=1$ [ANS]\n$-9x+9\\text{is an expression.}$ [ANS]\n$\\text{We can evaluate}-9x+9\\text{when}x=1$ [ANS]\n$9x-9\\text{is an equation.}$ [ANS]\n$\\text{We can check whether}x=1\\text{is a solution of}-9x+9=9x-9.$ [ANS]", "answer_v2": [ "True", "False", "False", "False", "True", "True", "False", "True" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v2": [ [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ] ], "problem_v3": "Choose True or False for the following questions about the difference between expressions and equations.\n$-4x+2=2x-4\\text{is an expression.}$ [ANS]\n$\\text{We can evaluate}-4x+2=2x-4\\text{when}x=1$ [ANS]\n$-4x+2=2x-4\\text{is an equation.}$ [ANS]\n$\\text{We can check whether}x=1\\text{is a solution of}-4x+2=2x-4.$ [ANS]\n$\\text{We can check whether}x=1\\text{is a solution of}-4x+2.$ [ANS]\n$-4x+2\\text{is an expression.}$ [ANS]\n$\\text{We can evaluate}-4x+2\\text{when}x=1$ [ANS]\n$2x-4\\text{is an equation.}$ [ANS]", "answer_v3": [ "False", "False", "True", "True", "False", "True", "True", "False" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v3": [ [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ] ] }, { "id": "Algebra_0003", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Properties", "level": "2", "keywords": [ "algebra" ], "problem_v1": "Match the statements defined below with the letters labeling particular numbers. Use all the letters. Of course a natural number is also a rational number, for example. However, there is only one correct matching that uses all five letters A through E. [ANS] 1. $x$ is neither positive nor negative [ANS] 2. $x$ is an irrational number [ANS] 3. $x$ is a rational number [ANS] 4. $x$ is an integer [ANS] 5. $x$ is a natural number\nA. $x=\\pi$ B. $x=-17$ C. $x=\\frac{17}{12}$ D. $x=0$ E. $x=12$", "answer_v1": [ "D", "A", "C", "B", "E" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Match the statements defined below with the letters labeling particular numbers. Use all the letters. Of course a natural number is also a rational number, for example. However, there is only one correct matching that uses all five letters A through E. [ANS] 1. $x$ is a natural number [ANS] 2. $x$ is an integer [ANS] 3. $x$ is a rational number [ANS] 4. $x$ is an irrational number [ANS] 5. $x$ is neither positive nor negative\nA. $x=-17$ B. $x=12$ C. $x=\\frac{17}{12}$ D. $x=0$ E. $x=\\pi$", "answer_v2": [ "B", "A", "C", "E", "D" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Match the statements defined below with the letters labeling particular numbers. Use all the letters. Of course a natural number is also a rational number, for example. However, there is only one correct matching that uses all five letters A through E. [ANS] 1. $x$ is a rational number [ANS] 2. $x$ is neither positive nor negative [ANS] 3. $x$ is a natural number [ANS] 4. $x$ is an integer [ANS] 5. $x$ is an irrational number\nA. $x=0$ B. $x=\\pi$ C. $x=-17$ D. $x=12$ E. $x=\\frac{17}{12}$", "answer_v3": [ "E", "A", "D", "C", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Algebra_0004", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Properties", "level": "1", "keywords": [ "algebra" ], "problem_v1": "Match the verbs below with the letters labeling particular symbols. [ANS] 1. divide [ANS] 2. subtract [ANS] 3. multiply [ANS] 4. add\nA. $-$ B. $/$ C. $+$ D. $\\times$", "answer_v1": [ "B", "A", "D", "C" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Match the verbs below with the letters labeling particular symbols. [ANS] 1. add [ANS] 2. divide [ANS] 3. subtract [ANS] 4. multiply\nA. $\\times$ B. $+$ C. $/$ D. $-$", "answer_v2": [ "B", "C", "D", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Match the verbs below with the letters labeling particular symbols. [ANS] 1. subtract [ANS] 2. multiply [ANS] 3. add [ANS] 4. divide\nA. $-$ B. $+$ C. $/$ D. $\\times$", "answer_v3": [ "A", "D", "B", "C" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Algebra_0005", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Properties", "level": "2", "keywords": [ "true-false" ], "problem_v1": "Indicate whether the following statements are True (T) or False (F). [ANS] 1. The difference of two real numbers is always an irrational number. [ANS] 2. The quotient of two real numbers is always a rational number (provided the denominator is non-zero). [ANS] 3. The quotient of two real numbers is always a real number (provided the denominator is non-zero). [ANS] 4. The product of two real numbers is always a real number. [ANS] 5. The sum of two real numbers is always a real number. [ANS] 6. The difference of two real numbers is always a real number. [ANS] 7. The ratio of two real numbers is never zero.", "answer_v1": [ "F", "F", "T", "T", "T", "T", "F" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "Indicate whether the following statements are True (T) or False (F). [ANS] 1. The sum of two real numbers is always a real number. [ANS] 2. The ratio of two real numbers is never zero. [ANS] 3. The difference of two real numbers is always a real number. [ANS] 4. The quotient of two real numbers is always a rational number (provided the denominator is non-zero). [ANS] 5. The difference of two real numbers is always an irrational number. [ANS] 6. The product of two real numbers is always a real number. [ANS] 7. The quotient of two real numbers is always a real number (provided the denominator is non-zero).", "answer_v2": [ "T", "F", "T", "F", "F", "T", "T" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "Indicate whether the following statements are True (T) or False (F). [ANS] 1. The product of two real numbers is always a real number. [ANS] 2. The quotient of two real numbers is always a real number (provided the denominator is non-zero). [ANS] 3. The difference of two real numbers is always a real number. [ANS] 4. The difference of two real numbers is always an irrational number. [ANS] 5. The sum of two real numbers is always a real number. [ANS] 6. The quotient of two real numbers is always a rational number (provided the denominator is non-zero). [ANS] 7. The ratio of two real numbers is never zero.", "answer_v3": [ "T", "T", "T", "F", "T", "F", "F" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Algebra_0006", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Properties", "level": "1", "keywords": [], "problem_v1": "$(a+b)+c=$ [ANS] A. $a+(b+c)$ B. $-a-b-c$ C. $a+b-c$ D. $ac+ab$\nWhich property does this expression demonstrate? [ANS] A. Additive Identity B. Multiplicative Inverse C. Associativity D. Additive Inverse E. Commutativity F. Multiplicative Identity", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "$(a+b)+c=$ [ANS] A. $a+(b+c)$ B. $-a-b-c$ C. $a+b-c$ D. $ac+ab$\nWhich property does this expression demonstrate? [ANS] A. Associativity B. Multiplicative Inverse C. Multiplicative Identity D. Commutativity E. Additive Identity F. Additive Inverse", "answer_v2": [ "A", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "$(a+b)+c=$ [ANS] A. $a+(b+c)$ B. $a+b-c$ C. $-a-b-c$ D. $ac+ab$\nWhich property does this expression demonstrate? [ANS] A. Multiplicative Identity B. Multiplicative Inverse C. Commutativity D. Associativity E. Additive Identity F. Additive Inverse", "answer_v3": [ "A", "D" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Algebra_0007", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Properties", "level": "2", "keywords": [ "Equations" ], "problem_v1": "Which of the following expressions are equivalent to $-8 (x-6)$? There may be more than one correct answer. [ANS] A. $8 (6-x)$ B. $-8x+6$ C. $-8x+48$ D. $48-8x$ E. $-8x-6$", "answer_v1": [ "ACD" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Which of the following expressions are equivalent to $-2 (x-9)$? There may be more than one correct answer. [ANS] A. $-2x+18$ B. $-2x+9$ C. $18-2x$ D. $-2x-9$ E. $2 (9-x)$", "answer_v2": [ "ACE" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Which of the following expressions are equivalent to $-4 (x-6)$? There may be more than one correct answer. [ANS] A. $-4x+24$ B. $24-4x$ C. $4 (6-x)$ D. $-4x-6$ E. $-4x+6$", "answer_v3": [ "ABC" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E" ] ] }, { "id": "Algebra_0008", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Properties", "level": "2", "keywords": [ "real numbers" ], "problem_v1": "List the elements of the set\n\\lbrace x | x \\mbox{is a natural number between 6 and 12} \\rbrace. Separate multiple answers by commas. Answer: [ANS]", "answer_v1": [ "(7, 8, 9, 10, 11)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "List the elements of the set\n\\lbrace x | x \\mbox{is a whole number between 7 and 10} \\rbrace. Separate multiple answers by commas. Answer: [ANS]", "answer_v2": [ "(8, 9)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "List the elements of the set\n\\lbrace x | x \\mbox{is a whole number between 6 and 10} \\rbrace. Separate multiple answers by commas. Answer: [ANS]", "answer_v3": [ "(7, 8, 9)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Algebra_0009", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Properties", "level": "2", "keywords": [ "prealgebra", "common core" ], "problem_v1": "Fill in the missing blank and determine the appropriate property illustrated:\n$1 \\times$ [ANS] $=\\frac{22+x}{y}$, uses [ANS]", "answer_v1": [ "(22+x)/y", "Identity Property of Multiplication" ], "answer_type_v1": [ "EX", "MCS" ], "options_v1": [ [], [ "Identity Property of Addition", "Identity Property of Multiplication", "Commutative Property of Addition", "Commutative Property of Multiplication", "Associative Property of Addition", "Associative Property of Multiplication" ] ], "problem_v2": "Fill in the missing blank and determine the appropriate property illustrated:\n$1 \\times$ [ANS] $=\\frac{15+x}{y}$, uses [ANS]", "answer_v2": [ "(15+x)/y", "Identity Property of Multiplication" ], "answer_type_v2": [ "EX", "MCS" ], "options_v2": [ [], [ "Identity Property of Addition", "Identity Property of Multiplication", "Commutative Property of Addition", "Commutative Property of Multiplication", "Associative Property of Addition", "Associative Property of Multiplication" ] ], "problem_v3": "Fill in the missing blank and determine the appropriate property illustrated:\n$1 \\times$ [ANS] $=\\frac{16+x}{y}$, uses [ANS]", "answer_v3": [ "(16+x)/y", "Identity Property of Multiplication" ], "answer_type_v3": [ "EX", "MCS" ], "options_v3": [ [], [ "Identity Property of Addition", "Identity Property of Multiplication", "Commutative Property of Addition", "Commutative Property of Multiplication", "Associative Property of Addition", "Associative Property of Multiplication" ] ] }, { "id": "Algebra_0010", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Algebraic expressions", "level": "5", "keywords": [], "problem_v1": "The number of students enrolled in math courses at Portland Community College has grown over the years. The formulas\n$\\begin{aligned}M&=0.53x+4.8 & \\qquad&& W&=0.49x+5.2\\end{aligned}$ describe the numbers (of thousands) of men and women enrolled in math courses at PCC $x$ years after 2005. Give a simplified formula for the total number $T$ of thousands of students at PCC taking math classes $x$ years after 2005. Be sure to give the entire formula, starting with $T=$. [ANS]", "answer_v1": [ "T = 1.02*x+10" ], "answer_type_v1": [ "EQ" ], "options_v1": [ [] ], "problem_v2": "The number of students enrolled in math courses at Portland Community College has grown over the years. The formulas\n$\\begin{aligned}M&=0.32x+5.8 & \\qquad&& W&=0.34x+4\\end{aligned}$ describe the numbers (of thousands) of men and women enrolled in math courses at PCC $x$ years after 2005. Give a simplified formula for the total number $T$ of thousands of students at PCC taking math classes $x$ years after 2005. Be sure to give the entire formula, starting with $T=$. [ANS]", "answer_v2": [ "T = 0.66*x+9.8" ], "answer_type_v2": [ "EQ" ], "options_v2": [ [] ], "problem_v3": "The number of students enrolled in math courses at Portland Community College has grown over the years. The formulas\n$\\begin{aligned}M&=0.39x+4.8 & \\qquad&& W&=0.38x+4.7\\end{aligned}$ describe the numbers (of thousands) of men and women enrolled in math courses at PCC $x$ years after 2005. Give a simplified formula for the total number $T$ of thousands of students at PCC taking math classes $x$ years after 2005. Be sure to give the entire formula, starting with $T=$. [ANS]", "answer_v3": [ "T = 0.77*x+9.5" ], "answer_type_v3": [ "EQ" ], "options_v3": [ [] ] }, { "id": "Algebra_0011", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Algebraic expressions", "level": "2", "keywords": [ "expression", "english-to-math" ], "problem_v1": "Translate the following phrase into a math expression or equation (whichever is appropriate). Use $x$ to represent the unknown number.\neight less than the quotient of seven and a number [ANS]", "answer_v1": [ "7/x-8" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Translate the following phrase into a math expression or equation (whichever is appropriate). Use $x$ to represent the unknown number.\none more than the quotient of ten and a number [ANS]", "answer_v2": [ "10/x+1" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Translate the following phrase into a math expression or equation (whichever is appropriate). Use $x$ to represent the unknown number.\nfour more than the quotient of seven and a number [ANS]", "answer_v3": [ "7/x+4" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0012", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Algebraic expressions", "level": "2", "keywords": [ "solve", "linear", "equation", "english-to-math", "subtract", "divide" ], "problem_v1": "Write an equation for the following situation, and then solve for the unknown. Please use $x$ as the unknown variable. The sum of fourteen and four times a number is forty-two. What is the number? The sum of fourteen and four times a number is forty-two. What is the number? Your equation is: [ANS]\nThe unknown number is: [ANS]", "answer_v1": [ "4*x = 28", "7" ], "answer_type_v1": [ "EQ", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Write an equation for the following situation, and then solve for the unknown. Please use $x$ as the unknown variable. The sum of ten and five times a number is twenty-five. What is the number? The sum of ten and five times a number is twenty-five. What is the number? Your equation is: [ANS]\nThe unknown number is: [ANS]", "answer_v2": [ "5*x = 15", "3" ], "answer_type_v2": [ "EQ", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Write an equation for the following situation, and then solve for the unknown. Please use $x$ as the unknown variable. The sum of eleven and four times a number is twenty-seven. What is the number? The sum of eleven and four times a number is twenty-seven. What is the number? Your equation is: [ANS]\nThe unknown number is: [ANS]", "answer_v3": [ "4*x = 16", "4" ], "answer_type_v3": [ "EQ", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0013", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Algebraic expressions", "level": "3", "keywords": [ "algebra", "modeling", "equations" ], "problem_v1": "The distance (in miles) traveled when driving at a certain speed $s$ for $39$ hours, then driving $14$ miles/hour faster for another hour. Express the distance in terms of $s$.\nYour answer is: [ANS]", "answer_v1": [ "(39+1)*s+14" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "The distance (in miles) traveled when driving at a certain speed $s$ for $8$ hours, then driving $19$ miles/hour faster for another hour. Express the distance in terms of $s$.\nYour answer is: [ANS]", "answer_v2": [ "(8+1)*s+19" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "The distance (in miles) traveled when driving at a certain speed $s$ for $19$ hours, then driving $14$ miles/hour faster for another hour. Express the distance in terms of $s$.\nYour answer is: [ANS]", "answer_v3": [ "(19+1)*s+14" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0014", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Algebraic expressions", "level": "2", "keywords": [ "algebra", "modeling", "equations" ], "problem_v1": "Express the average age of three sisters in the terms of the age $a$ of the firstborn (in years) if the second was born $5$ years after the first and the third was born $4$ years after the second.\nYour answer is: [ANS]", "answer_v1": [ "(3*a-2*5-4)/3" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Express the average age of three sisters in the terms of the age $a$ of the firstborn (in years) if the second was born $2$ years after the first and the third was born $6$ years after the second.\nYour answer is: [ANS]", "answer_v2": [ "(3*a-2*2-6)/3" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Express the average age of three sisters in the terms of the age $a$ of the firstborn (in years) if the second was born $3$ years after the first and the third was born $5$ years after the second.\nYour answer is: [ANS]", "answer_v3": [ "(3*a-2*3-5)/3" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0015", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Algebraic expressions", "level": "3", "keywords": [ "algebra", "modeling", "equations" ], "problem_v1": "Express the sum of $5$ consecutive integers in terms of the first integer $n$ of them.\nYour answer is: [ANS]", "answer_v1": [ "5*n+(5*(5-1))/2" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Express the sum of $2$ consecutive integers in terms of the first integer $n$ of them.\nYour answer is: [ANS]", "answer_v2": [ "2*n+(2*(2-1))/2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Express the sum of $3$ consecutive integers in terms of the first integer $n$ of them.\nYour answer is: [ANS]", "answer_v3": [ "3*n+(3*(3-1))/2" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0016", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Algebraic expressions", "level": "2", "keywords": [ "algebra" ], "problem_v1": "The basic idea of manipulating algebraic expressions is that they obey the same laws as arithmetic expressions. The following are some simple exercises along those lines. They ask you to enter numerical values for the variables $A$, $B$, $C$... The expression $6(5-5x)$ equals $Ax+B$ where $A$ equals: [ANS]\nand $B$ equals: [ANS]\n[NOTE: Your answers cannot be algebraic expressions.]", "answer_v1": [ "-30", "30" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The basic idea of manipulating algebraic expressions is that they obey the same laws as arithmetic expressions. The following are some simple exercises along those lines. They ask you to enter numerical values for the variables $A$, $B$, $C$... The expression $2(7-2x)$ equals $Ax+B$ where $A$ equals: [ANS]\nand $B$ equals: [ANS]\n[NOTE: Your answers cannot be algebraic expressions.]", "answer_v2": [ "-4", "14" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The basic idea of manipulating algebraic expressions is that they obey the same laws as arithmetic expressions. The following are some simple exercises along those lines. They ask you to enter numerical values for the variables $A$, $B$, $C$... The expression $3(5-3x)$ equals $Ax+B$ where $A$ equals: [ANS]\nand $B$ equals: [ANS]\n[NOTE: Your answers cannot be algebraic expressions.]", "answer_v3": [ "-9", "15" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0017", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Algebraic expressions", "level": "2", "keywords": [ "Algebraic fractions" ], "problem_v1": "Write the expression as a single fraction. Simplify your answer.\n$\\begin{array}{cccc}\\hline & \\frac{6}{a}+\\frac{5}{b}=& & [ANS] [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "6*b+5*a", "a*b" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Write the expression as a single fraction. Simplify your answer.\n$\\begin{array}{cccc}\\hline & \\frac{3}{a}+\\frac{7}{b}=& & [ANS] [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "3*b+7*a", "a*b" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Write the expression as a single fraction. Simplify your answer.\n$\\begin{array}{cccc}\\hline & \\frac{4}{a}+\\frac{6}{b}=& & [ANS] [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "4*b+6*a", "a*b" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0018", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Algebraic expressions", "level": "2", "keywords": [ "Expressions" ], "problem_v1": "(a) Write an expression for the total cost of buying 9 apples at \\$ $a$ each and 4 pears at \\$ $p$ each. Your expression should be in terms of $a$ and $p$. \\$ [ANS]\n(b) Find the total cost if apples cost \\$ $0.50$ each and pears cost \\$ $0.85$ each. \\$ [ANS]", "answer_v1": [ "9*a+4*p", "7.9" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "(a) Write an expression for the total cost of buying 6 apples at \\$ $a$ each and 5 pears at \\$ $p$ each. Your expression should be in terms of $a$ and $p$. \\$ [ANS]\n(b) Find the total cost if apples cost \\$ $0.40$ each and pears cost \\$ $0.75$ each. \\$ [ANS]", "answer_v2": [ "6*a+5*p", "6.15" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "(a) Write an expression for the total cost of buying 7 apples at \\$ $a$ each and 4 pears at \\$ $p$ each. Your expression should be in terms of $a$ and $p$. \\$ [ANS]\n(b) Find the total cost if apples cost \\$ $0.45$ each and pears cost \\$ $0.80$ each. \\$ [ANS]", "answer_v3": [ "7*a+4*p", "6.35" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0019", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Algebraic expressions", "level": "2", "keywords": [ "Expressions" ], "problem_v1": "(a) Suppose you pick two numbers 5 and 6. Find their sum and product, but do not enter it below. Then, find the average of their sum and product and enter it below. [ANS]\n(b) Using the variables $x$ and $y$ to stand for the two numbers, write an algebraic expression that represents this calculation. [ANS]", "answer_v1": [ "20.5", "(x+y+x*y)/2" ], "answer_type_v1": [ "NV", "EX" ], "options_v1": [ [], [] ], "problem_v2": "(a) Suppose you pick two numbers 2 and 9. Find their sum and product, but do not enter it below. Then, find the average of their sum and product and enter it below. [ANS]\n(b) Using the variables $x$ and $y$ to stand for the two numbers, write an algebraic expression that represents this calculation. [ANS]", "answer_v2": [ "14.5", "(x+y+x*y)/2" ], "answer_type_v2": [ "NV", "EX" ], "options_v2": [ [], [] ], "problem_v3": "(a) Suppose you pick two numbers 3 and 6. Find their sum and product, but do not enter it below. Then, find the average of their sum and product and enter it below. [ANS]\n(b) Using the variables $x$ and $y$ to stand for the two numbers, write an algebraic expression that represents this calculation. [ANS]", "answer_v3": [ "13.5", "(x+y+x*y)/2" ], "answer_type_v3": [ "NV", "EX" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0020", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Algebraic expressions", "level": "2", "keywords": [ "Expressions" ], "problem_v1": "A caterer for a party buys 75 cans of soda and 20 bags of chips. Write an expression for the total cost if soda costs $s$ dollars per can and chips cost $c$ dollars per bag.\n\\$ [ANS]", "answer_v1": [ "75*s+20*c" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "A caterer for a party buys 60 cans of soda and 25 bags of chips. Write an expression for the total cost if soda costs $s$ dollars per can and chips cost $c$ dollars per bag.\n\\$ [ANS]", "answer_v2": [ "60*s+25*c" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "A caterer for a party buys 65 cans of soda and 20 bags of chips. Write an expression for the total cost if soda costs $s$ dollars per can and chips cost $c$ dollars per bag.\n\\$ [ANS]", "answer_v3": [ "65*s+20*c" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0021", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Algebraic expressions", "level": "2", "keywords": [ "Equations" ], "problem_v1": "Determine whether the expressions are equivalent or not.\n[ANS] 1. $-a+2$ and $-(a-2)$\n[ANS] 2. $-a+2$ and $-2a+4/2$\n[ANS] 3. $-a+2$ and $2-a$\n[ANS] 4. $-a+2$ and $-(2-a)$\n[ANS] 5. $-a+2$ and $-(a+2)$", "answer_v1": [ "EQUIVALENT", "NOT EQUIVALENT", "EQUIVALENT", "Not Equivalent", "Not Equivalent" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "Equivalent", "Not Equivalent" ], [ "Equivalent", "Not Equivalent" ], [ "Equivalent", "Not Equivalent" ], [ "Equivalent", "Not Equivalent" ], [ "Equivalent", "Not Equivalent" ] ], "problem_v2": "Determine whether the expressions are equivalent or not.\n[ANS] 1. $-a+2$ and $-(2-a)$\n[ANS] 2. $-a+2$ and $-2a+4/2$\n[ANS] 3. $-a+2$ and $-(a+2)$\n[ANS] 4. $-a+2$ and $-(a-2)$\n[ANS] 5. $-a+2$ and $2-a$", "answer_v2": [ "NOT EQUIVALENT", "NOT EQUIVALENT", "NOT EQUIVALENT", "Equivalent", "Equivalent" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "Equivalent", "Not Equivalent" ], [ "Equivalent", "Not Equivalent" ], [ "Equivalent", "Not Equivalent" ], [ "Equivalent", "Not Equivalent" ], [ "Equivalent", "Not Equivalent" ] ], "problem_v3": "Determine whether the expressions are equivalent or not.\n[ANS] 1. $-a+2$ and $2-a$\n[ANS] 2. $-a+2$ and $-2a+4/2$\n[ANS] 3. $-a+2$ and $-(a+2)$\n[ANS] 4. $-a+2$ and $-(2-a)$\n[ANS] 5. $-a+2$ and $-(a-2)$", "answer_v3": [ "EQUIVALENT", "NOT EQUIVALENT", "NOT EQUIVALENT", "Not Equivalent", "Equivalent" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "Equivalent", "Not Equivalent" ], [ "Equivalent", "Not Equivalent" ], [ "Equivalent", "Not Equivalent" ], [ "Equivalent", "Not Equivalent" ], [ "Equivalent", "Not Equivalent" ] ] }, { "id": "Algebra_0022", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Algebraic expressions", "level": "2", "keywords": [ "Equations" ], "problem_v1": "Write the expression $(8x)(7x)+(6x) (8x)+7 (6x)+x(6x)$ in a simpler form, if possible. If it is not possible to simplify, re-write the given expression. [ANS]", "answer_v1": [ "110*x^2+42*x" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Write the expression $(2x)(3x)+(9x) (2x)+3 (9x)+x(9x)$ in a simpler form, if possible. If it is not possible to simplify, re-write the given expression. [ANS]", "answer_v2": [ "33*x^2+27*x" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Write the expression $(4x)(3x)+(6x) (4x)+3 (6x)+x(6x)$ in a simpler form, if possible. If it is not possible to simplify, re-write the given expression. [ANS]", "answer_v3": [ "42*x^2+18*x" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0023", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Algebraic expressions", "level": "2", "keywords": [ "real numbers" ], "problem_v1": "Two angles are supplementary if the sum of their measures is $180^{\\circ}$. If the measure of one angle is $7x$ degrees, represent the measure of its supplement as an expression of $x$. Answer: [ANS]", "answer_v1": [ "180-7*x" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Two angles are complementary if the sum of their measures is $90^{\\circ}$. If the measure of an angle is $10x$ degrees, represent the measure of its complement as an expression in $x$. Answer: [ANS]", "answer_v2": [ "90-10*x" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Two angles are complementary if the sum of their measures is $90^{\\circ}$. If the measure of an angle is $12x$ degrees, represent the measure of its complement as an expression in $x$. Answer: [ANS]", "answer_v3": [ "90-12*x" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0024", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Algebraic expressions", "level": "2", "keywords": [ "real numbers" ], "problem_v1": "Two numbers have a sum of $153$. If one number is $k$, represent the other number as an expression of $k$. Answer: [ANS]", "answer_v1": [ "153-k" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Two numbers have a sum of $239$. If one number is $x$, represent the other number as an expression of $x$. Answer: [ANS]", "answer_v2": [ "239-x" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Two numbers have a sum of $158$. If one number is $p$, represent the other number as an expression of $p$. Answer: [ANS]", "answer_v3": [ "158-p" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0025", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Algebraic expressions", "level": "1", "keywords": [ "prealgebra", "common core" ], "problem_v1": "Fill in the blanks below by choosing from the available options Deductive Reasoning: The process of reasoning logically to get a conclusion from given [ANS] Simplify: The process of replacing an expression with an equivalent one that has a smaller number of [ANS] Constant: A term without a [ANS] Like Terms: Terms which have the same [ANS] Term: A number or the product of a number and a [ANS] Coefficient: In each term, the variable is multiplied by this [ANS]", "answer_v1": [ "Facts", "Terms", "Variable", "Variable", "Variable", "Number" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "Variable", "Terms", "Facts", "Number" ], [ "Variable", "Terms", "Facts", "Number" ], [ "Variable", "Terms", "Facts", "Number" ], [ "Variable", "Terms", "Facts", "Number" ], [ "Variable", "Terms", "Facts", "Number" ], [ "Variable", "Terms", "Facts", "Number" ] ], "problem_v2": "Fill in the blanks below by choosing from the available options Term: A number or the product of a number and a [ANS] Like Terms: Terms which have the same [ANS] Simplify: The process of replacing an expression with an equivalent one that has a smaller number of [ANS] Coefficient: In each term, the variable is multiplied by this [ANS] Deductive Reasoning: The process of reasoning logically to get a conclusion from given [ANS] Constant: A term without a [ANS]", "answer_v2": [ "Variable", "Variable", "Terms", "Number", "Facts", "Variable" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "Variable", "Terms", "Facts", "Number" ], [ "Variable", "Terms", "Facts", "Number" ], [ "Variable", "Terms", "Facts", "Number" ], [ "Variable", "Terms", "Facts", "Number" ], [ "Variable", "Terms", "Facts", "Number" ], [ "Variable", "Terms", "Facts", "Number" ] ], "problem_v3": "Fill in the blanks below by choosing from the available options Deductive Reasoning: The process of reasoning logically to get a conclusion from given [ANS] Term: A number or the product of a number and a [ANS] Like Terms: Terms which have the same [ANS] Coefficient: In each term, the variable is multiplied by this [ANS] Constant: A term without a [ANS] Simplify: The process of replacing an expression with an equivalent one that has a smaller number of [ANS]", "answer_v3": [ "Facts", "Variable", "Variable", "Number", "Variable", "Terms" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "Variable", "Terms", "Facts", "Number" ], [ "Variable", "Terms", "Facts", "Number" ], [ "Variable", "Terms", "Facts", "Number" ], [ "Variable", "Terms", "Facts", "Number" ], [ "Variable", "Terms", "Facts", "Number" ], [ "Variable", "Terms", "Facts", "Number" ] ] }, { "id": "Algebra_0026", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Algebraic expressions", "level": "2", "keywords": [ "prealgebra", "common core" ], "problem_v1": "Write a variable expression for the phrase\n$15$ subtracted from a number $k$ [ANS]", "answer_v1": [ "k-15" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Write a variable expression for the phrase\n$3$ subtracted from a number $k$ [ANS]", "answer_v2": [ "k-3" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Write a variable expression for the phrase\n$7$ subtracted from a number $k$ [ANS]", "answer_v3": [ "k-7" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0027", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Algebraic expressions", "level": "2", "keywords": [ "prealgebra", "common core" ], "problem_v1": "Write variable expressions for\nThe value in cents of $a$ nickels=[ANS]\nThe value in cents of $a$ dimes=[ANS]\nThe value in cents of $a$ quarters=[ANS]\nThe value in cents of $a$ dimes and one quarter=[ANS]\nThe value in cents of $17$ dimes=[ANS]", "answer_v1": [ "a*5", "a*10", "a*25", "a*10+25", "17*10" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Write variable expressions for\nThe value in cents of $a$ nickels=[ANS]\nThe value in cents of $a$ dimes=[ANS]\nThe value in cents of $a$ quarters=[ANS]\nThe value in cents of $a$ dimes and one quarter=[ANS]\nThe value in cents of $11$ dimes=[ANS]", "answer_v2": [ "a*5", "a*10", "a*25", "a*10+25", "11*10" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Write variable expressions for\nThe value in cents of $a$ nickels=[ANS]\nThe value in cents of $a$ dimes=[ANS]\nThe value in cents of $a$ quarters=[ANS]\nThe value in cents of $a$ dimes and one quarter=[ANS]\nThe value in cents of $13$ dimes=[ANS]", "answer_v3": [ "a*5", "a*10", "a*25", "a*10+25", "13*10" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Algebra_0028", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Algebraic expressions", "level": "1", "keywords": [ "prealgebra", "common core" ], "problem_v1": "Write a variable expression for the phrase\nThe product of $21$ and a number $p$ [ANS]", "answer_v1": [ "p*21" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Write a variable expression for the phrase\nThe product of $12$ and a number $p$ [ANS]", "answer_v2": [ "p*12" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Write a variable expression for the phrase\nThe product of $15$ and a number $p$ [ANS]", "answer_v3": [ "p*15" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0029", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Algebraic expressions", "level": "4", "keywords": [ "prealgebra", "common core", "exponents" ], "problem_v1": "Write the following using exponential notation only (do not simplify):\nsix to the fifth power=[ANS]\nfour to the sixth power=[ANS] Use the ^ symbol for powers. Use the ^ symbol for powers.", "answer_v1": [ "6^5", "4^6" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Write the following using exponential notation only (do not simplify):\ntwo to the second power=[ANS]\nseven to the fourth power=[ANS] Use the ^ symbol for powers. Use the ^ symbol for powers.", "answer_v2": [ "2^2", "7^4" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Write the following using exponential notation only (do not simplify):\nthree to the third power=[ANS]\nsix to the fifth power=[ANS] Use the ^ symbol for powers. Use the ^ symbol for powers.", "answer_v3": [ "3^3", "6^5" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0031", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Evaluating expressions", "level": "2", "keywords": [ "financial mathematics", "algebra" ], "problem_v1": "Evaluate the expression $\\left(\\frac{4}{-3} \\right)^4$. [ANS]\n[NOTE: Your answer cannot be an algebraic expression.]", "answer_v1": [ "3.16049382716049" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the expression $\\left(\\frac{2}{-2} \\right)^2$. [ANS]\n[NOTE: Your answer cannot be an algebraic expression.]", "answer_v2": [ "1" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the expression $\\left(\\frac{2}{-3} \\right)^3$. [ANS]\n[NOTE: Your answer cannot be an algebraic expression.]", "answer_v3": [ "-0.296296296296296" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0032", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Evaluating expressions", "level": "3", "keywords": [ "integer", "negative", "operation", "add", "subtract", "evaluate" ], "problem_v1": "Evaluate this expression:\n${\\sqrt{12^2+5^2}=}$ [ANS]", "answer_v1": [ "13" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate this expression:\n${\\sqrt{4^2+3^2}=}$ [ANS]", "answer_v2": [ "5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate this expression:\n${\\sqrt{8^2+6^2}=}$ [ANS]", "answer_v3": [ "10" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0033", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Evaluating expressions", "level": "2", "keywords": [ "evaluate", "multivariable", "polynomial" ], "problem_v1": "Evaluate the following expression for $x=2$ and $y=1$\n${xy^{2}+2xy-2}$ The expression evaluates to [ANS].", "answer_v1": [ "4" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the following expression for $x=-3$ and $y=3$\n${-4xy^{2}-2xy+5}$ The expression evaluates to [ANS].", "answer_v2": [ "131" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the following expression for $x=-1$ and $y=1$\n${-2xy^{2}+xy-3}$ The expression evaluates to [ANS].", "answer_v3": [ "-2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0034", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Evaluating expressions", "level": "1", "keywords": [ "evaluate", "rational", "fraction", "multivariable" ], "problem_v1": "A formula for converting miles into kilometers is\n${K=1.61M}$ where $M$ is a number of miles, and $K$ is the corresponding number of kilometers. Use the formula to find the number of kilometers that corresponds to thirteen miles. [ANS] kilometers corresponds to thirteen miles.", "answer_v1": [ "20.93" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A formula for converting pounds into kilograms is\n${K=0.45P}$ where $P$ is a number of pounds, and $K$ is the corresponding number of kilograms. Use the formula to find the number of kilograms that corresponds to nineteen pounds. [ANS] kilograms corresponds to nineteen pounds.", "answer_v2": [ "8.55" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A formula for converting meters into feet is\n${F=3.28M}$ where $M$ is a number of meters, and $F$ is the corresponding number of feet. Use the formula to find the number of feet that corresponds to thirteen meters. [ANS] feet corresponds to thirteen meters.", "answer_v3": [ "42.64" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0036", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Evaluating expressions", "level": "3", "keywords": [ "evaluate", "expression", "fraction" ], "problem_v1": "To convert a temperature measured in degrees Fahrenheit to degrees Celsius, there is a formula:\n${C={\\frac{5}{9}\\!\\left(F-32\\right)}}$ where $C$ represents the temperature in degrees Celsius and $F$ represents the temperature in degrees Fahrenheit. If a temperature is $95 {^\\circ}\\text{F}$, what is that temperature measured in Celsius? Use degC for $^{\\circ}\\text{C}$ and degF for $^{\\circ}\\text{F}$. [ANS]", "answer_v1": [ "35" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "To convert a temperature measured in degrees Fahrenheit to degrees Celsius, there is a formula:\n${C={\\frac{5}{9}\\!\\left(F-32\\right)}}$ where $C$ represents the temperature in degrees Celsius and $F$ represents the temperature in degrees Fahrenheit. If a temperature is $14 {^\\circ}\\text{F}$, what is that temperature measured in Celsius? Use degC for $^{\\circ}\\text{C}$ and degF for $^{\\circ}\\text{F}$. [ANS]", "answer_v2": [ "-10" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "To convert a temperature measured in degrees Fahrenheit to degrees Celsius, there is a formula:\n${C={\\frac{5}{9}\\!\\left(F-32\\right)}}$ where $C$ represents the temperature in degrees Celsius and $F$ represents the temperature in degrees Fahrenheit. If a temperature is $41 {^\\circ}\\text{F}$, what is that temperature measured in Celsius? Use degC for $^{\\circ}\\text{C}$ and degF for $^{\\circ}\\text{F}$. [ANS]", "answer_v3": [ "5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0037", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Evaluating expressions", "level": "3", "keywords": [ "evaluate", "rational", "fraction", "multivariable" ], "problem_v1": "The diagonal length ($D$) of a rectangle with side lengths $L$ and $W$ is given by:\n${D=\\sqrt{L^2+W^2}}$ Determine the diagonal length of rectangles with $L={7\\ {\\rm ft}}$ and $W={24\\ {\\rm ft}}$. The diagonal length of rectangles with $L={7\\ {\\rm ft}}$ and $W={24\\ {\\rm ft}}$ is [ANS].", "answer_v1": [ "25" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The diagonal length ($D$) of a rectangle with side lengths $L$ and $W$ is given by:\n${D=\\sqrt{L^2+W^2}}$ Determine the diagonal length of rectangles with $L={3\\ {\\rm ft}}$ and $W={4\\ {\\rm ft}}$. The diagonal length of rectangles with $L={3\\ {\\rm ft}}$ and $W={4\\ {\\rm ft}}$ is [ANS].", "answer_v2": [ "5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The diagonal length ($D$) of a rectangle with side lengths $L$ and $W$ is given by:\n${D=\\sqrt{L^2+W^2}}$ Determine the diagonal length of rectangles with $L={9\\ {\\rm ft}}$ and $W={12\\ {\\rm ft}}$. The diagonal length of rectangles with $L={9\\ {\\rm ft}}$ and $W={12\\ {\\rm ft}}$ is [ANS].", "answer_v3": [ "15" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0038", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Evaluating expressions", "level": "3", "keywords": [ "evaluate", "linear", "integer" ], "problem_v1": "Evaluate the following expressions.\nEvaluate ${2t^{2}}$ when $t=3$. ${{2t^{2}}=}$ [ANS]\nEvaluate ${\\left(2t\\right)^{2}}$ when $t=3$. ${{\\left(2t\\right)^{2}}=}$ [ANS]", "answer_v1": [ "18", "36" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Evaluate the following expressions.\nEvaluate ${5x^{2}}$ when $x=2$. ${{5x^{2}}=}$ [ANS]\nEvaluate ${\\left(5x\\right)^{2}}$ when $x=2$. ${{\\left(5x\\right)^{2}}=}$ [ANS]", "answer_v2": [ "20", "100" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Evaluate the following expressions.\nEvaluate ${4y^{2}}$ when $y=2$. ${{4y^{2}}=}$ [ANS]\nEvaluate ${\\left(4y\\right)^{2}}$ when $y=2$. ${{\\left(4y\\right)^{2}}=}$ [ANS]", "answer_v3": [ "16", "64" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0039", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Evaluating expressions", "level": "3", "keywords": [ "evaluate", "rational", "fraction", "multivariable" ], "problem_v1": "The height inside a camping tent when you are $d$ feet from the edge of the tent is given by\n${h={-\\left|d-5\\right|+5}}$ where $h$ stands for height in feet. Determine the height when you are:\n${7.5\\ {\\rm ft}}$ from the edge. The height inside a camping tent when you ${7.5\\ {\\rm ft}}$ from the edge of the tent is [ANS]\n${2.5\\ {\\rm ft}}$ from the edge. The height inside a camping tent when you ${2.5\\ {\\rm ft}}$ from the edge of the tent is [ANS]", "answer_v1": [ "2.5", "2.5" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The height inside a camping tent when you are $d$ feet from the edge of the tent is given by\n${h={-\\left|d-4.2\\right|+7}}$ where $h$ stands for height in feet. Determine the height when you are:\n${7.3\\ {\\rm ft}}$ from the edge. The height inside a camping tent when you ${7.3\\ {\\rm ft}}$ from the edge of the tent is [ANS]\n${3.3\\ {\\rm ft}}$ from the edge. The height inside a camping tent when you ${3.3\\ {\\rm ft}}$ from the edge of the tent is [ANS]", "answer_v2": [ "3.9 ft", "6.1 ft" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The height inside a camping tent when you are $d$ feet from the edge of the tent is given by\n${h={-1.3\\!\\left|d-5\\right|+3.5}}$ where $h$ stands for height in feet. Determine the height when you are:\n${7.6\\ {\\rm ft}}$ from the edge. The height inside a camping tent when you ${7.6\\ {\\rm ft}}$ from the edge of the tent is [ANS]\n${3.5\\ {\\rm ft}}$ from the edge. The height inside a camping tent when you ${3.5\\ {\\rm ft}}$ from the edge of the tent is [ANS]", "answer_v3": [ "0.12 ft", "1.55 ft" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0040", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Evaluating expressions", "level": "3", "keywords": [ "evaluate", "rational", "fraction", "multivariable" ], "problem_v1": "The percentage of births in the U.S. delivered via C-section can be given by the following formula for the years since 1996.\n${p=0.8(y-1996)+21}$ In this formula $y$ is a year after 1996 and $p$ is the percentage of births delivered via C-section for that year. What percentage of births in the U.S. were delivered via C-section in the year 2010? [ANS]% of births in the U.S. were delivered via C-section in the year 2010.", "answer_v1": [ "32.2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The percentage of births in the U.S. delivered via C-section can be given by the following formula for the years since 1996.\n${p=0.8(y-1996)+21}$ In this formula $y$ is a year after 1996 and $p$ is the percentage of births delivered via C-section for that year. What percentage of births in the U.S. were delivered via C-section in the year 1999? [ANS]% of births in the U.S. were delivered via C-section in the year 1999.", "answer_v2": [ "23.4" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The percentage of births in the U.S. delivered via C-section can be given by the following formula for the years since 1996.\n${p=0.8(y-1996)+21}$ In this formula $y$ is a year after 1996 and $p$ is the percentage of births delivered via C-section for that year. What percentage of births in the U.S. were delivered via C-section in the year 2003? [ANS]% of births in the U.S. were delivered via C-section in the year 2003.", "answer_v3": [ "26.6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0041", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Evaluating expressions", "level": "1", "keywords": [ "evaluate", "linear", "integer", "multivariable" ], "problem_v1": "Evaluate the expression for $B=3$ and $A=5$.\n${-3B+7A}=$ [ANS]", "answer_v1": [ "26" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the expression for $a=-7$ and $C=-3$.\n${-9a-2C}=$ [ANS]", "answer_v2": [ "69" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the expression for $b=-5$ and $A=1$.\n${-2b+9A}=$ [ANS]", "answer_v3": [ "19" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0042", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Evaluating expressions", "level": "3", "keywords": [ "formula", "expression", "evaluate" ], "problem_v1": "In your answers to this problem, use m for meters, and s for seconds. The formula\n${y=\\frac{1}{2}\\,a\\,t^2+v_0\\,t+y_0}$ gives the vertical position of an object, at time $t$, thrown with an initial velocity $v_0$, from an initial position $y_0$ in a place where the acceleration of gravity is $a$. What is the height of a baseball thrown with an initial velocity of $v_0={88\\ {\\textstyle\\frac{\\rm\\mathstrut m}{\\rm\\mathstrut s}}}$, from an initial position of $y_0={79\\ {\\rm m}}$, and at time $t={12\\ {\\rm s}}$? Note that the acceleration of gravity on earth is ${-9.8\\ {\\textstyle\\frac{\\rm\\mathstrut m}{\\rm\\mathstrut s^{2}}}}$. It is negative, because we consider the upward direction as positive in this situation, while the gravity pulls the object down.\nAfter ${12\\ {\\rm s}}$, the baseball was [ANS] high in the air.", "answer_v1": [ "429.4" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "In your answers to this problem, use m for meters, and s for seconds. The formula\n${y=\\frac{1}{2}\\,a\\,t^2+v_0\\,t+y_0}$ gives the vertical position of an object, at time $t$, thrown with an initial velocity $v_0$, from an initial position $y_0$ in a place where the acceleration of gravity is $a$. What is the height of a baseball thrown with an initial velocity of $v_0={54\\ {\\textstyle\\frac{\\rm\\mathstrut m}{\\rm\\mathstrut s}}}$, from an initial position of $y_0={97\\ {\\rm m}}$, and at time $t={2\\ {\\rm s}}$? Note that the acceleration of gravity on earth is ${-9.8\\ {\\textstyle\\frac{\\rm\\mathstrut m}{\\rm\\mathstrut s^{2}}}}$. It is negative, because we consider the upward direction as positive in this situation, while the gravity pulls the object down.\nAfter ${2\\ {\\rm s}}$, the baseball was [ANS] high in the air.", "answer_v2": [ "185.4" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "In your answers to this problem, use m for meters, and s for seconds. The formula\n${y=\\frac{1}{2}\\,a\\,t^2+v_0\\,t+y_0}$ gives the vertical position of an object, at time $t$, thrown with an initial velocity $v_0$, from an initial position $y_0$ in a place where the acceleration of gravity is $a$. What is the height of a baseball thrown with an initial velocity of $v_0={65\\ {\\textstyle\\frac{\\rm\\mathstrut m}{\\rm\\mathstrut s}}}$, from an initial position of $y_0={80\\ {\\rm m}}$, and at time $t={4\\ {\\rm s}}$? Note that the acceleration of gravity on earth is ${-9.8\\ {\\textstyle\\frac{\\rm\\mathstrut m}{\\rm\\mathstrut s^{2}}}}$. It is negative, because we consider the upward direction as positive in this situation, while the gravity pulls the object down.\nAfter ${4\\ {\\rm s}}$, the baseball was [ANS] high in the air.", "answer_v3": [ "261.6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0043", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Evaluating expressions", "level": "1", "keywords": [ "Algebra", "Celsius/Farenheit" ], "problem_v1": "Your friend from Paris arrives in New York and the forecast is for a low of 53 and a high of 77 degrees Fahrenheit. What is the forecasted low temperature in Celsius? [ANS]\nWhat is the foceasted high temperature in Celsius? [ANS]", "answer_v1": [ "5/9*(77-32)", "5/9*(53-32)" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Your friend from Paris arrives in New York and the forecast is for a low of 45 and a high of 74 degrees Fahrenheit. What is the forecasted low temperature in Celsius? [ANS]\nWhat is the foceasted high temperature in Celsius? [ANS]", "answer_v2": [ "5/9*(74-32)", "5/9*(45-32)" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Your friend from Paris arrives in New York and the forecast is for a low of 48 and a high of 72 degrees Fahrenheit. What is the forecasted low temperature in Celsius? [ANS]\nWhat is the foceasted high temperature in Celsius? [ANS]", "answer_v3": [ "5/9*(72-32)", "5/9*(48-32)" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0044", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Evaluating expressions", "level": "2", "keywords": [ "algebra" ], "problem_v1": "Evaluate the expression $|-(51-158)|$. Answer: [ANS]", "answer_v1": [ "107" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the expression $|-(-84-194)|$. Answer: [ANS]", "answer_v2": [ "278" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the expression $|-(-37-161)|$. Answer: [ANS]", "answer_v3": [ "198" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0045", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Evaluating expressions", "level": "2", "keywords": [ "real numbers", "operations with signed numbers" ], "problem_v1": "Evaluate each exponential expression: a) $(-1)^{60}=$ [ANS]\nb) $-(-\\frac{1}{3})^3=$ [ANS]\n(Note: Your answer must be a fraction.)", "answer_v1": [ "1", "1/27" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Evaluate each exponential expression: a) $(-1)^{20}=$ [ANS]\nb) $-(-\\frac{4}{5})^2=$ [ANS]\n(Note: Your answer must be a fraction.)", "answer_v2": [ "1", "-16/25" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Evaluate each exponential expression: a) $(-1)^{34}=$ [ANS]\nb) $-(-\\frac{2}{3})^3=$ [ANS]\n(Note: Your answer must be a fraction.)", "answer_v3": [ "1", "8/27" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0046", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Evaluating expressions", "level": "2", "keywords": [], "problem_v1": "A construction company must pay a fine for completing a job late. The company uses the equation below to calculate the amount of the fine $f$ in dollars when the job is finished $d$ days late. f=20,000+1,500 d. The company completes a construction job $3$ days late. The fine is \\$ [ANS].", "answer_v1": [ "24500" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A construction company must pay a fine for completing a job late. The company uses the equation below to calculate the amount of the fine $f$ in dollars when the job is finished $d$ days late. f=25,000+1,200 d. The company completes a construction job $3$ days late. The fine is \\$ [ANS].", "answer_v2": [ "28600" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A construction company must pay a fine for completing a job late. The company uses the equation below to calculate the amount of the fine $f$ in dollars when the job is finished $d$ days late. f=20,000+1,300 d. The company completes a construction job $3$ days late. The fine is \\$ [ANS].", "answer_v3": [ "23900" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0047", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Evaluating expressions", "level": "3", "keywords": [], "problem_v1": "The manager of a restaurant uses the formula below to decide what to charge for a meal.\np=(f\\div 3)\\times 10 In the formula, $p$ is the price, in dollars. that customers pay for a meal and $f$ is the food cost to make the meal. If the food cost is \\$10.50 then the price of the meal is \\$ [ANS].", "answer_v1": [ "35" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The manager of a restaurant uses the formula below to decide what to charge for a meal.\np=(f\\div 3)\\times 10 In the formula, $p$ is the price, in dollars. that customers pay for a meal and $f$ is the food cost to make the meal. If the food cost is \\$5.25 then the price of the meal is \\$ [ANS].", "answer_v2": [ "17.50" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The manager of a restaurant uses the formula below to decide what to charge for a meal.\np=(f\\div 3)\\times 10 In the formula, $p$ is the price, in dollars. that customers pay for a meal and $f$ is the food cost to make the meal. If the food cost is \\$4.50 then the price of the meal is \\$ [ANS].", "answer_v3": [ "15" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0048", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Evaluating expressions", "level": "2", "keywords": [], "problem_v1": "What is the value of $\\frac{3}{4} x-\\frac{1}{2}$ when $x=24$? [ANS] A. $17 \\frac{1}{2}$ B. $31 \\frac{1}{2}$ C. $8$ D. $4$", "answer_v1": [ "A" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "What is the value of $\\frac{2}{3} x-\\frac{1}{2}$ when $x=18$? [ANS] A. $6$ B. $26 \\frac{1}{2}$ C. $3$ D. $11 \\frac{1}{2}$", "answer_v2": [ "D" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "What is the value of $\\frac{2}{3} x-\\frac{1}{2}$ when $x=18$? [ANS] A. $6$ B. $11 \\frac{1}{2}$ C. $26 \\frac{1}{2}$ D. $3$", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Algebra_0049", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Evaluating expressions", "level": "2", "keywords": [], "problem_v1": "Look at this inequality. $\\square > \\vert-16 \\vert$ Which expression makes the inequality true? [ANS] A. $\\vert-17 \\vert$ B. $-\\vert 17 \\vert$ C. $-\\vert 15 \\vert$ D. $\\vert-15 \\vert$", "answer_v1": [ "A" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "Look at this inequality. $\\square > \\vert-12 \\vert$ Which expression makes the inequality true? [ANS] A. $-\\vert 11 \\vert$ B. $-\\vert 13 \\vert$ C. $\\vert-11 \\vert$ D. $\\vert-13 \\vert$", "answer_v2": [ "D" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "Look at this inequality. $\\square > \\vert-13 \\vert$ Which expression makes the inequality true? [ANS] A. $-\\vert 12 \\vert$ B. $\\vert-14 \\vert$ C. $-\\vert 14 \\vert$ D. $\\vert-12 \\vert$", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Algebra_0050", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Evaluating expressions", "level": "2", "keywords": [], "problem_v1": "Evaluate each of the following expressions. Your answer must be in simplest form [a proper fraction or mixed number] If $p={\\textstyle\\frac{1}{7}}$ and $q=3 \\frac{4}{7}$ then $p^2q=$ [ANS]\nIf $s={\\textstyle\\frac{1}{7}}$ and $t=1 \\frac{2}{7}$ then $t^2/s=$ [ANS]", "answer_v1": [ "25/343", "11 4/7" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Evaluate each of the following expressions. Your answer must be in simplest form [a proper fraction or mixed number] If $p={\\textstyle\\frac{1}{3}}$ and $q=5 \\frac{1}{7}$ then $p^2q=$ [ANS]\nIf $s={\\textstyle\\frac{1}{5}}$ and $t=2 \\frac{2}{7}$ then $t^2/s=$ [ANS]", "answer_v2": [ "4/7", "26 6/49" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Evaluate each of the following expressions. Your answer must be in simplest form [a proper fraction or mixed number] If $p={\\textstyle\\frac{1}{3}}$ and $q=4 \\frac{2}{7}$ then $p^2q=$ [ANS]\nIf $s={\\textstyle\\frac{1}{5}}$ and $t=1 \\frac{3}{7}$ then $t^2/s=$ [ANS]", "answer_v3": [ "10/21", "10 10/49" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0051", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Evaluating expressions", "level": "2", "keywords": [ "algebra" ], "problem_v1": "Note that in general $a^b$ does not equal $a\\times b$. Evaluate the following arithmetic expressions and enter them as an integer. $4^2-4 \\times 2=$ [ANS]. $6^2-6 \\times 2=$ [ANS].", "answer_v1": [ "8", "24" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Note that in general $a^b$ does not equal $a\\times b$. Evaluate the following arithmetic expressions and enter them as an integer. $2^2-2 \\times 2=$ [ANS]. $5^2-5 \\times 2=$ [ANS].", "answer_v2": [ "0", "15" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Note that in general $a^b$ does not equal $a\\times b$. Evaluate the following arithmetic expressions and enter them as an integer. $2^2-2 \\times 2=$ [ANS]. $4^2-4 \\times 2=$ [ANS].", "answer_v3": [ "0", "8" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0052", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Evaluating expressions", "level": "2", "keywords": [ "Expressions" ], "problem_v1": "Evaluate the expression $ \\frac{1}{2} h \\big(B+b \\big)$ when $h=12, \\ B=6, \\ b=7.$ [ANS]", "answer_v1": [ "78" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the expression $ \\frac{1}{2} h \\big(B+b \\big)$ when $h=6, \\ B=8, \\ b=3.$ [ANS]", "answer_v2": [ "33" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the expression $ \\frac{1}{2} h \\big(B+b \\big)$ when $h=8, \\ B=6, \\ b=4.$ [ANS]", "answer_v3": [ "40" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0053", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Evaluating expressions", "level": "2", "keywords": [ "Equations" ], "problem_v1": "Answer yes or no.\n[ANS] 1. Is $t=0$ a solution to the equation $ t+1=\\frac{1-t}{t}$?\n[ANS] 2. Is $a=0$ a solution to the equation $ \\frac{3+a}{3-a}=1$?\n[ANS] 3. Is $t=0$ a solution to the equation $ 20-t=20+t$?\n[ANS] 4. Is $x=4$ a solution to the equation $ x+3=x^2-9$?\n[ANS] 5. Is $t=3$ a solution to the equation $ t+3=t^2+9$?", "answer_v1": [ "NO", "YES", "YES", "Yes", "No" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF" ], "options_v1": [ [ "Yes", "No" ], [ "Yes", "No" ], [ "Yes", "No" ], [ "Yes", "No" ], [ "Yes", "No" ] ], "problem_v2": "Answer yes or no.\n[ANS] 1. Is $x=4$ a solution to the equation $ x+3=x^2-9$?\n[ANS] 2. Is $a=0$ a solution to the equation $ \\frac{3+a}{3-a}=1$?\n[ANS] 3. Is $t=3$ a solution to the equation $ t+3=t^2+9$?\n[ANS] 4. Is $t=0$ a solution to the equation $ t+1=\\frac{1-t}{t}$?\n[ANS] 5. Is $t=0$ a solution to the equation $ 20-t=20+t$?", "answer_v2": [ "YES", "YES", "NO", "No", "Yes" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF" ], "options_v2": [ [ "Yes", "No" ], [ "Yes", "No" ], [ "Yes", "No" ], [ "Yes", "No" ], [ "Yes", "No" ] ], "problem_v3": "Answer yes or no.\n[ANS] 1. Is $t=0$ a solution to the equation $ 20-t=20+t$?\n[ANS] 2. Is $a=0$ a solution to the equation $ \\frac{3+a}{3-a}=1$?\n[ANS] 3. Is $t=3$ a solution to the equation $ t+3=t^2+9$?\n[ANS] 4. Is $x=4$ a solution to the equation $ x+3=x^2-9$?\n[ANS] 5. Is $t=0$ a solution to the equation $ t+1=\\frac{1-t}{t}$?", "answer_v3": [ "YES", "YES", "NO", "Yes", "No" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF" ], "options_v3": [ [ "Yes", "No" ], [ "Yes", "No" ], [ "Yes", "No" ], [ "Yes", "No" ], [ "Yes", "No" ] ] }, { "id": "Algebra_0054", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Evaluating expressions", "level": "2", "keywords": [ "Equations" ], "problem_v1": "If $x+y+z=27$, find the value of $(y-8)+(z+5)+(x-7).$ [ANS]", "answer_v1": [ "17" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If $x+y+z=23$, find the value of $(y-2)+(z+9)+(x-3).$ [ANS]", "answer_v2": [ "27" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If $x+y+z=26$, find the value of $(y-4)+(z+7)+(x-4).$ [ANS]", "answer_v3": [ "25" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0055", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Evaluating expressions", "level": "3", "keywords": [ "algebra", "radicals", "triangle" ], "problem_v1": "Find the area of an equilateral triangle, each of whose sides is $9$ inches long. Area (in square inches): [ANS]", "answer_v1": [ "sqrt(3)*9^2/4" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the area of an equilateral triangle, each of whose sides is $4$ inches long. Area (in square inches): [ANS]", "answer_v2": [ "sqrt(3)*4^2/4" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the area of an equilateral triangle, each of whose sides is $6$ inches long. Area (in square inches): [ANS]", "answer_v3": [ "sqrt(3)*6^2/4" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0056", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Inequalities and intervals", "level": "2", "keywords": [ "proof\u2019", "\u2018logic\u2019", "\u2018grammar\u2019", "\u2018convention\u2019", "interval notation" ], "problem_v1": "When expressing sets, you may write \u201cinf\u201d for infinity. Give the interval notation for $\\lbrace x: x < 5 \\rbrace$ [ANS]\nGive the interval notation for $\\lbrace x:-3 < x \\leq 5 \\rbrace$ [ANS]", "answer_v1": [ "(-infinity,5)", "(-3,5]" ], "answer_type_v1": [ "INT", "INT" ], "options_v1": [ [], [] ], "problem_v2": "When expressing sets, you may write \u201cinf\u201d for infinity. Give the interval notation for $\\lbrace x: x < 1 \\rbrace$ [ANS]\nGive the interval notation for $\\lbrace x:-1 < x \\leq 1 \\rbrace$ [ANS]", "answer_v2": [ "(-infinity,1)", "(-1,1]" ], "answer_type_v2": [ "INT", "INT" ], "options_v2": [ [], [] ], "problem_v3": "When expressing sets, you may write \u201cinf\u201d for infinity. Give the interval notation for $\\lbrace x: x < 2 \\rbrace$ [ANS]\nGive the interval notation for $\\lbrace x:-3 < x \\leq 2 \\rbrace$ [ANS]", "answer_v3": [ "(-infinity,2)", "(-3,2]" ], "answer_type_v3": [ "INT", "INT" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0057", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Inequalities and intervals", "level": "3", "keywords": [ "algebra", "interval" ], "problem_v1": "Let $S=(0,\\infty)$, $T=(-\\infty,4]$, and $W=[-4,4)$. For each intersection or union, choose the correct notation for the resulting interval. [ANS] 1. $S\\cup W$ [ANS] 2. $S\\cap W$ [ANS] 3. $T\\cap W$ [ANS] 4. $S\\cup T$\nA. $[-4,4)$ B. $(0,4)$ C. $[-4,\\infty)$ D. $(-\\infty,\\infty)$", "answer_v1": [ "C", "B", "A", "D" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Let $S=(-4,\\infty)$, $T=(-\\infty,-3]$, and $W=[-10,-3)$. For each intersection or union, choose the correct notation for the resulting interval. [ANS] 1. $T\\cap W$ [ANS] 2. $T\\cup W$ [ANS] 3. $S\\cap W$ [ANS] 4. $S\\cap T$\nA. $(-\\infty,-3]$ B. $(-4,-3)$ C. $[-10,-3)$ D. $(-4,-3]$", "answer_v2": [ "C", "A", "B", "D" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Let $S=(-4,\\infty)$, $T=(-\\infty,-2]$, and $W=[-8,-2)$. For each intersection or union, choose the correct notation for the resulting interval. [ANS] 1. $S\\cup T$ [ANS] 2. $S\\cap W$ [ANS] 3. $T\\cap W$ [ANS] 4. $T\\cup W$\nA. $(-\\infty,-2]$ B. $[-8,-2)$ C. $(-\\infty,\\infty)$ D. $(-4,-2)$", "answer_v3": [ "C", "D", "B", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Algebra_0058", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Inequalities and intervals", "level": "", "keywords": [ "inequalities" ], "problem_v1": "Solve the compound inequality. x \\leq 6 \\mbox{and} x \\leq 4 Answer: [ANS]", "answer_v1": [ "(-infinity, 4]" ], "answer_type_v1": [ "INT" ], "options_v1": [ [] ], "problem_v2": "Solve the compound inequality. x \\leq 0 \\mbox{and} x \\leq 8 Answer: [ANS]", "answer_v2": [ "(-infinity, 0]" ], "answer_type_v2": [ "INT" ], "options_v2": [ [] ], "problem_v3": "Solve the compound inequality. x \\leq 2 \\mbox{and} x \\leq 5 Answer: [ANS]", "answer_v3": [ "(-infinity, 2]" ], "answer_type_v3": [ "INT" ], "options_v3": [ [] ] }, { "id": "Algebra_0059", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Inequalities and intervals", "level": "2", "keywords": [ "algebra" ], "problem_v1": "Your department sends its copying to the photocopy center of your company. The center bills your department \\$ 0.11 per page. You have investigated the possibility of buying a departmental copier for \\$ 3800. With your own copier, the cost per page would be \\$ 0.04. The expected life of the copier is 6 years. You figure that you must make at least [ANS] copies per year to justify buying the copier. Your answer should be a natural number. (Of course this question totally ignores the convenience of having your own copier. Also, the number of copies is likely to go up, and so may the price charged by the copy center.) Hint: Set up and solve an inequality.", "answer_v1": [ "9047" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Your department sends its copying to the photocopy center of your company. The center bills your department \\$ 0.08 per page. You have investigated the possibility of buying a departmental copier for \\$ 4400. With your own copier, the cost per page would be \\$ 0.02. The expected life of the copier is 4 years. You figure that you must make at least [ANS] copies per year to justify buying the copier. Your answer should be a natural number. (Of course this question totally ignores the convenience of having your own copier. Also, the number of copies is likely to go up, and so may the price charged by the copy center.) Hint: Set up and solve an inequality.", "answer_v2": [ "18333" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Your department sends its copying to the photocopy center of your company. The center bills your department \\$ 0.09 per page. You have investigated the possibility of buying a departmental copier for \\$ 3800. With your own copier, the cost per page would be \\$ 0.03. The expected life of the copier is 5 years. You figure that you must make at least [ANS] copies per year to justify buying the copier. Your answer should be a natural number. (Of course this question totally ignores the convenience of having your own copier. Also, the number of copies is likely to go up, and so may the price charged by the copy center.) Hint: Set up and solve an inequality.", "answer_v3": [ "12666" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0060", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Inequalities and intervals", "level": "3", "keywords": [ "inequality" ], "problem_v1": "Match the with the given [ANS] 1. $[3,5)$ [ANS] 2. $(3,5]$ [ANS] 3. $(3,5)$ [ANS] 4. $[3,5]$\nA. $3 < x \\leq 5$ B. $3 \\leq x < 5$ C. $3 \\leq x \\leq 5$ D. $3 < x < 5$", "answer_v1": [ "B", "A", "D", "C" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Match the with the given [ANS] 1. $[3,5]$ [ANS] 2. $[3,5)$ [ANS] 3. $(3,5]$ [ANS] 4. $(3,5)$\nA. $3 < x < 5$ B. $3 \\leq x \\leq 5$ C. $3 \\leq x < 5$ D. $3 < x \\leq 5$", "answer_v2": [ "B", "C", "D", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Match the with the given [ANS] 1. $(3,5]$ [ANS] 2. $(3,5)$ [ANS] 3. $[3,5]$ [ANS] 4. $[3,5)$\nA. $3 < x \\leq 5$ B. $3 \\leq x \\leq 5$ C. $3 \\leq x < 5$ D. $3 < x < 5$", "answer_v3": [ "A", "D", "B", "C" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Algebra_0061", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Simplifying expressions", "level": "4", "keywords": [ "algebra", "powers" ], "problem_v1": "If you rationalize the numerator of \\frac{\\sqrt[3]{x^2}-5 \\sqrt[3]{x}+25}{\\sqrt{x^3}+5} then you will get \\frac{A}{B} where A=[ANS]\nand B=[ANS]", "answer_v1": [ "x + 5**3", "x**(11/6)+5*x**(1/3) +5*x^(3/2) + (5)**2" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "If you rationalize the numerator of \\frac{\\sqrt[3]{x^2}+8 \\sqrt[3]{x}+64}{\\sqrt{x^3}-8} then you will get \\frac{A}{B} where A=[ANS]\nand B=[ANS]", "answer_v2": [ "x + -8**3", "x**(11/6)+-8*x**(1/3) +-8*x^(3/2) + (-8)**2" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "If you rationalize the numerator of \\frac{\\sqrt[3]{x^2}+4 \\sqrt[3]{x}+16}{\\sqrt{x^3}-4} then you will get \\frac{A}{B} where A=[ANS]\nand B=[ANS]", "answer_v3": [ "x + -4**3", "x**(11/6)+-4*x**(1/3) +-4*x^(3/2) + (-4)**2" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0062", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Simplifying expressions", "level": "2", "keywords": [ "algebra", "operations with rational expressions" ], "problem_v1": "Simplify the expression 2-\\frac{4}{x-3} and give your answer in the form of \\frac{f(x)}{g(x)}. Your answer for the function $f(x)$ is: [ANS]\nYour answer for the function $g(x)$ is: [ANS]", "answer_v1": [ "2 x-4-2*3", "x-3" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Simplify the expression 3-\\frac{1}{x-1} and give your answer in the form of \\frac{f(x)}{g(x)}. Your answer for the function $f(x)$ is: [ANS]\nYour answer for the function $g(x)$ is: [ANS]", "answer_v2": [ "3 x-1-3*1", "x-1" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Simplify the expression 2-\\frac{2}{x-2} and give your answer in the form of \\frac{f(x)}{g(x)}. Your answer for the function $f(x)$ is: [ANS]\nYour answer for the function $g(x)$ is: [ANS]", "answer_v3": [ "2 x-2-2*2", "x-2" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0063", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Simplifying expressions", "level": "3", "keywords": [ "algebra", "fractions", "complex fractions" ], "problem_v1": "The total resistance, $R$, of a particular group is given by the formula: R=S+\\left(\\frac{1}{\\frac{1}{T}+\\frac{1}{W}}\\right)\nThis formula can be simplified to the form $\\frac{A}{B}$ where $A$ and $B$ contain no fractions. $A=$ [ANS]\n$B=$ [ANS]\nSuppose that $S=27 \\Omega$, $T=64 \\Omega$ and $W=95 \\Omega$. Then $R=$ [ANS] $\\Omega$ Note: Your answer must be a decimal.", "answer_v1": [ "T * W + S*(T+W)", "T+W", "65.2389937106918" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The total resistance, $R$, of a particular group is given by the formula: R=S+\\left(\\frac{1}{\\frac{1}{T}+\\frac{1}{W}}\\right)\nThis formula can be simplified to the form $\\frac{A}{B}$ where $A$ and $B$ contain no fractions. $A=$ [ANS]\n$B=$ [ANS]\nSuppose that $S=12 \\Omega$, $T=89 \\Omega$ and $W=86 \\Omega$. Then $R=$ [ANS] $\\Omega$ Note: Your answer must be a decimal.", "answer_v2": [ "T * W + S*(T+W)", "T+W", "55.7371428571429" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The total resistance, $R$, of a particular group is given by the formula: R=S+\\left(\\frac{1}{\\frac{1}{T}+\\frac{1}{W}}\\right)\nThis formula can be simplified to the form $\\frac{A}{B}$ where $A$ and $B$ contain no fractions. $A=$ [ANS]\n$B=$ [ANS]\nSuppose that $S=17 \\Omega$, $T=65 \\Omega$ and $W=89 \\Omega$. Then $R=$ [ANS] $\\Omega$ Note: Your answer must be a decimal.", "answer_v3": [ "T * W + S*(T+W)", "T+W", "54.5649350649351" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0064", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Simplifying expressions", "level": "3", "keywords": [ "rational expressions", "least common denominator" ], "problem_v1": "Find the least common denominator (LCD) of the rational expressions: \\frac{11}{15x^{2}-8x-12} \\mbox{and} \\frac{1}{3x+2} LCD: [ANS]", "answer_v1": [ "(5*x-6)*(3*x+2)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the least common denominator (LCD) of the rational expressions: \\frac{18}{4x^{2}-8x-21} \\mbox{and} \\frac{1}{2x+3} LCD: [ANS]", "answer_v2": [ "(2*x-7)*(2*x+3)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the least common denominator (LCD) of the rational expressions: \\frac{4}{9x^{2}-3x-20} \\mbox{and} \\frac{1}{3x+4} LCD: [ANS]", "answer_v3": [ "(3*x-5)*(3*x+4)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0065", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Simplifying expressions", "level": "2", "keywords": [ "rational expressions", "rational functions" ], "problem_v1": "$(y^2-9)$ Perform the indicated operation. Note that the denominators are the same. Simplify the result, if possible. \\frac{y^{2}+3y+3}{y^{2}+6y+9}-\\frac{12+3y}{y^{2}+6y+9} Answer: [ANS]", "answer_v1": [ "(y-3)/(y+3)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "$(y^2+4*y-5)$ Perform the indicated operation. Note that the denominators are the same. Simplify the result, if possible. \\frac{y^{2}+5y-2}{y^{2}-25}-\\frac{y+3}{y^{2}-25} Answer: [ANS]", "answer_v2": [ "(y-1)/(y-5)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "$(y^2+y-12)$ Perform the indicated operation. Note that the denominators are the same. Simplify the result, if possible. \\frac{y^{2}+4y-4}{y^{2}+2y-8}-\\frac{8+3y}{y^{2}+2y-8} Answer: [ANS]", "answer_v3": [ "(y-3)/(y-2)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0066", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Simplifying expressions", "level": "2", "keywords": [ "Equations" ], "problem_v1": "Use the distributive law to rewrite the expression $5 ab(4 a-6 b)$ as an equivalent expression with no parentheses. [ANS]", "answer_v1": [ "20*a^2*b-30*a*b^2" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Use the distributive law to rewrite the expression $2 ab(6 a-3 b)$ as an equivalent expression with no parentheses. [ANS]", "answer_v2": [ "12*a^2*b-6*a*b^2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Use the distributive law to rewrite the expression $3 ab(5 a-4 b)$ as an equivalent expression with no parentheses. [ANS]", "answer_v3": [ "15*a^2*b-12*a*b^2" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0067", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Simplifying expressions", "level": "2", "keywords": [ "functions" ], "problem_v1": "Put the function $ Q=\\frac{1}{5} t \\sqrt{7}$ in the form $Q=k t$ and state the value of $k$.\n$k$=[ANS]", "answer_v1": [ "[sqrt(7)]/5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Put the function $ Q=\\frac{1}{7} t \\sqrt{3}$ in the form $Q=k t$ and state the value of $k$.\n$k$=[ANS]", "answer_v2": [ "[sqrt(3)]/7" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Put the function $ Q=\\frac{1}{6} t \\sqrt{3}$ in the form $Q=k t$ and state the value of $k$.\n$k$=[ANS]", "answer_v3": [ "[sqrt(3)]/6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0068", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Simplifying expressions", "level": "2", "keywords": [ "linear expressions" ], "problem_v1": "Give the constant term and the coefficient of $x$ in the linear expression $10x+r x$.\nThe constant term is [ANS]\nThe coefficient of $x$ is [ANS]", "answer_v1": [ "0", "10+r" ], "answer_type_v1": [ "NV", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Give the constant term and the coefficient of $x$ in the linear expression $3x+r x$.\nThe constant term is [ANS]\nThe coefficient of $x$ is [ANS]", "answer_v2": [ "0", "3+r" ], "answer_type_v2": [ "NV", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Give the constant term and the coefficient of $x$ in the linear expression $6x+r x$.\nThe constant term is [ANS]\nThe coefficient of $x$ is [ANS]", "answer_v3": [ "0", "6+r" ], "answer_type_v3": [ "NV", "EX" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0069", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Simplifying expressions", "level": "2", "keywords": [ "prealgebra", "common core" ], "problem_v1": "Consider the expression $7 j-11 j+4 j$. The coefficients=[ANS] (separate by a comma) and $7 j-11 j+4 j=$ [ANS]", "answer_v1": [ "(7, -11, 4)", "0*j" ], "answer_type_v1": [ "OL", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Consider the expression $4 j-13 j+1 j$. The coefficients=[ANS] (separate by a comma) and $4 j-13 j+1 j=$ [ANS]", "answer_v2": [ "(4, -13, 1)", "-8*j" ], "answer_type_v2": [ "OL", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Consider the expression $5 j-12 j+2 j$. The coefficients=[ANS] (separate by a comma) and $5 j-12 j+2 j=$ [ANS]", "answer_v3": [ "(5, -12, 2)", "-5*j" ], "answer_type_v3": [ "OL", "EX" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0070", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Solving linear equations in one variable", "level": "2", "keywords": [ "algebra", "linear equations" ], "problem_v1": "Solve the equation $8x+2=3x+5$ algebraically.\n$x=$ [ANS]", "answer_v1": [ "0.6" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Solve the equation $5x+9=2x-3$ algebraically.\n$x=$ [ANS]", "answer_v2": [ "-4" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Solve the equation $6x+2=2x+2$ algebraically.\n$x=$ [ANS]", "answer_v3": [ "0" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0071", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Solving linear equations in one variable", "level": "2", "keywords": [], "problem_v1": "Ken wants to earn \\$52.50 to buy a computer game. He uses the equation below to determine the number of hours, $h$, he needs to babysit to earn the \\$52.50. $5h+15=52.5$. Ken needs to babysit for [ANS] hours to earn enough money to buy the computer game.", "answer_v1": [ "7.5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Ken wants to earn \\$42.50 to buy a computer game. He uses the equation below to determine the number of hours, $h$, he needs to babysit to earn the \\$42.50. $5h+20=42.5$. Ken needs to babysit for [ANS] hours to earn enough money to buy the computer game.", "answer_v2": [ "4.5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Ken wants to earn \\$42.50 to buy a computer game. He uses the equation below to determine the number of hours, $h$, he needs to babysit to earn the \\$42.50. $5h+15=42.5$. Ken needs to babysit for [ANS] hours to earn enough money to buy the computer game.", "answer_v3": [ "5.5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0072", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Solving linear equations in one variable", "level": "2", "keywords": [], "problem_v1": "Solve each equation: a) $q\\times 12=192 \\ \\ \\ \\ \\ \\ q=$ [ANS]. b) $r\\div 18=23\\ \\ \\ \\ \\ \\ r=$ [ANS].", "answer_v1": [ "16", "414" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Solve each equation: a) $q\\times 12=156 \\ \\ \\ \\ \\ \\ q=$ [ANS]. b) $r\\div 19=21\\ \\ \\ \\ \\ \\ r=$ [ANS].", "answer_v2": [ "13", "399" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Solve each equation: a) $q\\times 12=168 \\ \\ \\ \\ \\ \\ q=$ [ANS]. b) $r\\div 18=22\\ \\ \\ \\ \\ \\ r=$ [ANS].", "answer_v3": [ "14", "396" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0073", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Solving linear equations in one variable", "level": "1", "keywords": [ "Equations" ], "problem_v1": "Solve the equation $ \\frac{T}{6}=10$.\n$T=$ [ANS]", "answer_v1": [ "60" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Solve the equation $ \\frac{T}{3}=11$.\n$T=$ [ANS]", "answer_v2": [ "33" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Solve the equation $ \\frac{T}{4}=10$.\n$T=$ [ANS]", "answer_v3": [ "40" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0074", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Solving linear equations in one variable", "level": "3", "keywords": [ "Solving Equations" ], "problem_v1": "Solve the equation $-9+10 r=-4 r$.\n$r=$ [ANS]", "answer_v1": [ "9/14" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Solve the equation $-6+12 r=-2 r$.\n$r=$ [ANS]", "answer_v2": [ "3/7" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Solve the equation $-7+10 r=-3 r$.\n$r=$ [ANS]", "answer_v3": [ "7/13" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0076", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Isolating variables", "level": "3", "keywords": [ "algebra", "solve for variable' 'fraction", "Equality", "Solve" ], "problem_v1": "Solve for $a$: (x+4 a)^2+(y-5 b)^2=9 There are two solutions, $a_1$ and $a_2$, where $a_1 \\leq a_2$. $a_1=$ [ANS] $a_2=$ [ANS]", "answer_v1": [ "(-x - sqrt(9 - (y-5*b)**2))/(--4)", "(-x + sqrt(9 - (y-5*b)**2))/(--4)" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Solve for $a$: (x+9 a)^2+(y+9 b)^2=9 There are two solutions, $a_1$ and $a_2$, where $a_1 \\leq a_2$. $a_1=$ [ANS] $a_2=$ [ANS]", "answer_v2": [ "(-x - sqrt(9 - (y--9*b)**2))/(--9)", "(-x + sqrt(9 - (y--9*b)**2))/(--9)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Solve for $a$: (x+4 a)^2+(y+4 b)^2=25 There are two solutions, $a_1$ and $a_2$, where $a_1 \\leq a_2$. $a_1=$ [ANS] $a_2=$ [ANS]", "answer_v3": [ "(-x - sqrt(25 - (y--4*b)**2))/(--4)", "(-x + sqrt(25 - (y--4*b)**2))/(--4)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0077", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Isolating variables", "level": "1", "keywords": [ "Equations" ], "problem_v1": "(a) What operation on both sides of the equation $7x=56$ isolates the variable on one side? There may be more than one correct answer. [ANS] A. Multiplying by $7$ B. Multiplying by $1/7$ C. Adding $7$ D. Dividing by $7$ E. Dividing by $7x$ F. Dividing by $56$ G. Subtracting $7x$\n(b) Give the solution of the equation $7x=56$. $x=$ [ANS]", "answer_v1": [ "BD", "8" ], "answer_type_v1": [ "MCM", "NV" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G" ], [] ], "problem_v2": "(a) What operation on both sides of the equation $3x=12$ isolates the variable on one side? There may be more than one correct answer. [ANS] A. Dividing by $12$ B. Adding $3$ C. Multiplying by $1/3$ D. Subtracting $3x$ E. Multiplying by $3$ F. Dividing by $3$ G. Dividing by $3x$\n(b) Give the solution of the equation $3x=12$. $x=$ [ANS]", "answer_v2": [ "CF", "4" ], "answer_type_v2": [ "MCM", "NV" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G" ], [] ], "problem_v3": "(a) What operation on both sides of the equation $4x=20$ isolates the variable on one side? There may be more than one correct answer. [ANS] A. Dividing by $4$ B. Subtracting $4x$ C. Dividing by $4x$ D. Dividing by $20$ E. Adding $4$ F. Multiplying by $1/4$ G. Multiplying by $4$\n(b) Give the solution of the equation $4x=20$. $x=$ [ANS]", "answer_v3": [ "AF", "5" ], "answer_type_v3": [ "MCM", "NV" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G" ], [] ] }, { "id": "Algebra_0078", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Scientific notation", "level": "3", "keywords": [ "exponent", "scientific", "notation", "negative" ], "problem_v1": "Divide the following numbers, writing your answer in scientific notation.\n$\\frac{4.8\\times 10^{4}}{8\\times 10^{-3}}=$ [ANS]", "answer_v1": [ "6*10^6" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Divide the following numbers, writing your answer in scientific notation.\n$\\frac{1.8\\times 10^{2}}{2\\times 10^{-4}}=$ [ANS]", "answer_v2": [ "9*10^5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Divide the following numbers, writing your answer in scientific notation.\n$\\frac{2.4\\times 10^{3}}{4\\times 10^{-5}}=$ [ANS]", "answer_v3": [ "6*10^7" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0079", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Scientific notation", "level": "2", "keywords": [ "exponent", "scientific", "notation", "zero" ], "problem_v1": "Write the following number in decimal notation without using exponents.\n$7.78\\times 10^{0}=$ [ANS]", "answer_v1": [ "7.78" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Write the following number in decimal notation without using exponents.\n$1.74\\times 10^{0}=$ [ANS]", "answer_v2": [ "1.74" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Write the following number in decimal notation without using exponents.\n$3.82\\times 10^{0}=$ [ANS]", "answer_v3": [ "3.82" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0080", "subject": "Algebra", "topic": "Algebra of real numbers and simplifying expressions", "subtopic": "Scientific notation", "level": "2", "keywords": [ "Scientific Notation" ], "problem_v1": "The mass of one hydrogen atom is $1.67 \\times 10^{-24}$ gram. Find the mass of 80,000 hydrogen atoms. Express the answer in scientific notation. Answer=[ANS] grams", "answer_v1": [ "1.336 x 10^-19" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The mass of one hydrogen atom is $1.67 \\times 10^{-24}$ gram. Find the mass of 50,000 hydrogen atoms. Express the answer in scientific notation. Answer=[ANS] grams", "answer_v2": [ "8.35 x 10^-20" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The mass of one hydrogen atom is $1.67 \\times 10^{-24}$ gram. Find the mass of 60,000 hydrogen atoms. Express the answer in scientific notation. Answer=[ANS] grams", "answer_v3": [ "1.002 x 10^-19" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0081", "subject": "Algebra", "topic": "Absolute value expressions and functions", "subtopic": "Solving equations with absolute values", "level": "2", "keywords": [ "absolute value" ], "problem_v1": "Find the absolute value of the following numbers.\n${|{8}|=}$ [ANS]\n${|{-8}|=}$ [ANS]\n${-|{8}|=}$ [ANS]\n${-|{-8}|=}$ [ANS]", "answer_v1": [ "8", "8", "-8", "-8" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Find the absolute value of the following numbers.\n${|{2}|=}$ [ANS]\n${|{-2}|=}$ [ANS]\n${-|{2}|=}$ [ANS]\n${-|{-2}|=}$ [ANS]", "answer_v2": [ "2", "2", "-2", "-2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Find the absolute value of the following numbers.\n${|{4}|=}$ [ANS]\n${|{-4}|=}$ [ANS]\n${-|{4}|=}$ [ANS]\n${-|{-4}|=}$ [ANS]", "answer_v3": [ "4", "4", "-4", "-4" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Algebra_0082", "subject": "Algebra", "topic": "Absolute value expressions and functions", "subtopic": "Solving equations with absolute values", "level": "3", "keywords": [ "Solve", "Equation", "Absolute Value" ], "problem_v1": "Solve the following equation for $x$: x\\ |\\ x-9 |=40x+6\nList the four possible roots in increasing order. Below each possible root, enter ROOT if it is a root or EXTRANEOUS if it is an extraneous root. [ANS] $<$ [ANS] $<$ [ANS] $<$ [ANS] [ANS], [ANS], [ANS], [ANS]", "answer_v1": [ "-30.805227865014", "-0.194772134986033", "-0.122144504490262", "49.1221445044903", "root", "root", "extraneous", "root" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [], [], [], [], [ "EXTRANEOUS", "ROOT" ], [ "EXTRANEOUS", "ROOT" ], [ "EXTRANEOUS", "ROOT" ], [ "EXTRANEOUS", "ROOT" ] ], "problem_v2": "Solve the following equation for $x$: x\\ |\\ x-3 |=48x+2\nList the four possible roots in increasing order. Below each possible root, enter ROOT if it is a root or EXTRANEOUS if it is an extraneous root. [ANS] $<$ [ANS] $<$ [ANS] $<$ [ANS] [ANS], [ANS], [ANS], [ANS]", "answer_v2": [ "-44.9555115728856", "-0.0444884271143806", "-0.0391855782442683", "51.0391855782443", "root", "root", "extraneous", "root" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [], [], [], [], [ "EXTRANEOUS", "ROOT" ], [ "EXTRANEOUS", "ROOT" ], [ "EXTRANEOUS", "ROOT" ], [ "EXTRANEOUS", "ROOT" ] ], "problem_v3": "Solve the following equation for $x$: x\\ |\\ x-5 |=36x+3\nList the four possible roots in increasing order. Below each possible root, enter ROOT if it is a root or EXTRANEOUS if it is an extraneous root. [ANS] $<$ [ANS] $<$ [ANS] $<$ [ANS] [ANS], [ANS], [ANS], [ANS]", "answer_v3": [ "-30.9029218007494", "-0.0970781992506371", "-0.0730406114409838", "41.073040611441", "root", "root", "extraneous", "root" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [], [], [], [], [ "EXTRANEOUS", "ROOT" ], [ "EXTRANEOUS", "ROOT" ], [ "EXTRANEOUS", "ROOT" ], [ "EXTRANEOUS", "ROOT" ] ] }, { "id": "Algebra_0083", "subject": "Algebra", "topic": "Absolute value expressions and functions", "subtopic": "Solving equations with absolute values", "level": "2", "keywords": [ "algebra", "equations", "Solve", "Equation", "Absolute Value", "Distance" ], "problem_v1": "The equation $|8x+32|=32$ has two solutions.\nThe distance between those two solutions is [ANS].", "answer_v1": [ "8" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The equation $|2x+12|=2$ has two solutions.\nThe distance between those two solutions is [ANS].", "answer_v2": [ "2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The equation $|4x+16|=8$ has two solutions.\nThe distance between those two solutions is [ANS].", "answer_v3": [ "4" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0084", "subject": "Algebra", "topic": "Absolute value expressions and functions", "subtopic": "Solving equations with absolute values", "level": "2", "keywords": [ "algebra", "expression", "absolute value" ], "problem_v1": "Use the properties of absolute value to simplify the expression |\\frac{1}{8} x-\\frac{6}{8}| to the form of A |x-B| The number $A$ is [ANS]\nThe number $B$ is [ANS].", "answer_v1": [ "0.125", "6" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Use the properties of absolute value to simplify the expression |\\frac{1}{2} x-\\frac{9}{2}| to the form of A |x-B| The number $A$ is [ANS]\nThe number $B$ is [ANS].", "answer_v2": [ "0.5", "9" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Use the properties of absolute value to simplify the expression |\\frac{1}{4} x-\\frac{6}{4}| to the form of A |x-B| The number $A$ is [ANS]\nThe number $B$ is [ANS].", "answer_v3": [ "0.25", "6" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0085", "subject": "Algebra", "topic": "Absolute value expressions and functions", "subtopic": "Solving equations with absolute values", "level": "2", "keywords": [ "algebra", "equation with absolute sign", "real zero" ], "problem_v1": "Find all real zeros of the equation $|12x|=10$. Its real zeros are $x_1=$ [ANS] and $x_2=$ [ANS] with $x_1\\le x_2$", "answer_v1": [ "-0.833333333333333", "0.833333333333333" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find all real zeros of the equation $|3x|=15$. Its real zeros are $x_1=$ [ANS] and $x_2=$ [ANS] with $x_1\\le x_2$", "answer_v2": [ "-5", "5" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find all real zeros of the equation $|6x|=10$. Its real zeros are $x_1=$ [ANS] and $x_2=$ [ANS] with $x_1\\le x_2$", "answer_v3": [ "-1.66666666666667", "1.66666666666667" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0086", "subject": "Algebra", "topic": "Absolute value expressions and functions", "subtopic": "Solving equations with absolute values", "level": "2", "keywords": [ "calculus", "algebra" ], "problem_v1": "Evaluate the expression $\\frac{|176-308|}{|-13|}$. Give you answer in decimal notation correct to three decimal places or give your answer as a fraction. [ANS]\nNOTE: You may not use absolute values in your answer.", "answer_v1": [ "132/13" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the expression $\\frac{|108-344|}{|-27|}$. Give you answer in decimal notation correct to three decimal places or give your answer as a fraction. [ANS]\nNOTE: You may not use absolute values in your answer.", "answer_v2": [ "236/27" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the expression $\\frac{|131-311|}{|-23|}$. Give you answer in decimal notation correct to three decimal places or give your answer as a fraction. [ANS]\nNOTE: You may not use absolute values in your answer.", "answer_v3": [ "180/23" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0087", "subject": "Algebra", "topic": "Absolute value expressions and functions", "subtopic": "Solving equations with absolute values", "level": "4", "keywords": [ "algebra", "equation", "absolute value" ], "problem_v1": "The equation $|x|=|y|$ is satisfied if $x=y$ or $x=-y$. Use this fact to solve the following equation. |x+6 |=| x-3 | Answer: $x=$ [ANS]\nHint: There is only one solution. There is only one solution.", "answer_v1": [ "-3/2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The equation $|x|=|y|$ is satisfied if $x=y$ or $x=-y$. Use this fact to solve the following equation. |x+2 |=| x-1 | Answer: $x=$ [ANS]\nHint: There is only one solution. There is only one solution.", "answer_v2": [ "-1/2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The equation $|x|=|y|$ is satisfied if $x=y$ or $x=-y$. Use this fact to solve the following equation. |x+2 |=| x-3 | Answer: $x=$ [ANS]\nHint: There is only one solution. There is only one solution.", "answer_v3": [ "1/2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0088", "subject": "Algebra", "topic": "Absolute value expressions and functions", "subtopic": "Absolute value inequalities", "level": "3", "keywords": [ "calculus", "intervals", "inequalities" ], "problem_v1": "Express the set of numbers $x$ satisfying the given condition as an interval:\n|7x-3|\\le 10 [ANS]", "answer_v1": [ "[-1,1.85714]" ], "answer_type_v1": [ "INT" ], "options_v1": [ [] ], "problem_v2": "Express the set of numbers $x$ satisfying the given condition as an interval:\n|4x-5|\\le 3 [ANS]", "answer_v2": [ "[0.5,2]" ], "answer_type_v2": [ "INT" ], "options_v2": [ [] ], "problem_v3": "Express the set of numbers $x$ satisfying the given condition as an interval:\n|5x-4|\\le 6 [ANS]", "answer_v3": [ "[-0.4,2]" ], "answer_type_v3": [ "INT" ], "options_v3": [ [] ] }, { "id": "Algebra_0090", "subject": "Algebra", "topic": "Absolute value expressions and functions", "subtopic": "Absolute value inequalities", "level": "2", "keywords": [ "Algebra", "Inequalities", "inequalities" ], "problem_v1": "To say that $|x-8 | \\leq 5$ is the same as saying $x$ is in the closed interval $[A,B]$ where $A$ is: [ANS]\nand where $B$ is: [ANS]", "answer_v1": [ "3", "13" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "To say that $|x-5 | \\leq 7$ is the same as saying $x$ is in the closed interval $[A,B]$ where $A$ is: [ANS]\nand where $B$ is: [ANS]", "answer_v2": [ "-2", "12" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "To say that $|x-6 | \\leq 5$ is the same as saying $x$ is in the closed interval $[A,B]$ where $A$ is: [ANS]\nand where $B$ is: [ANS]", "answer_v3": [ "1", "11" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0092", "subject": "Algebra", "topic": "Absolute value expressions and functions", "subtopic": "Applications using absolute values", "level": "2", "keywords": [], "problem_v1": "Find the distance betweem $333$ and $401$. [ANS]\n[NOTE: Your answer can be an algebraic expression]", "answer_v1": [ "68" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the distance betweem $473$ and $133$. [ANS]\n[NOTE: Your answer can be an algebraic expression]", "answer_v2": [ "340" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the distance betweem $342$ and $225$. [ANS]\n[NOTE: Your answer can be an algebraic expression]", "answer_v3": [ "117" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0093", "subject": "Algebra", "topic": "Absolute value expressions and functions", "subtopic": "Applications using absolute values", "level": "2", "keywords": [ "absolute value", "distance" ], "problem_v1": "Find the distance between the pair of real numbers: ${\\textstyle\\frac{7}{12}}$, ${\\textstyle\\frac{3}{10}}$. Answer: [ANS]", "answer_v1": [ "17/60" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the distance between the pair of real numbers: $-{\\textstyle\\frac{2}{3}}$, ${\\textstyle\\frac{2}{15}}$. Answer: [ANS]", "answer_v2": [ "4/5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the distance between the pair of real numbers: ${\\textstyle\\frac{5}{6}}$, $-{\\textstyle\\frac{7}{10}}$. Answer: [ANS]", "answer_v3": [ "23/15" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0094", "subject": "Algebra", "topic": "Properties of exponents, rational exponents and radicals", "subtopic": "Properties of exponents", "level": "2", "keywords": [ "financial mathematics", "algebra", "algebra", "powers", "calculus" ], "problem_v1": "The expression $(3a^5 b^4 c^4)^2(2a^4 b^3 c^3)^3$ equals $na^rb^sc^t$ where $n$, the leading coefficient, is: [ANS]\nand $r$, the exponent of $a$, is: [ANS]\nand $s$, the exponent of $b$, is: [ANS]\nand finally $t$, the exponent of $c$, is: [ANS]\n[NOTE: Your answers cannot be algebraic expressions.]", "answer_v1": [ "72", "22", "17", "17" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "The expression $(3a^2 b^5 c^2)^2(2a^3 b^5 c^3)^3$ equals $na^rb^sc^t$ where $n$, the leading coefficient, is: [ANS]\nand $r$, the exponent of $a$, is: [ANS]\nand $s$, the exponent of $b$, is: [ANS]\nand finally $t$, the exponent of $c$, is: [ANS]\n[NOTE: Your answers cannot be algebraic expressions.]", "answer_v2": [ "72", "13", "25", "13" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "The expression $(3a^3 b^4 c^3)^2(2a^4 b^2 c^3)^3$ equals $na^rb^sc^t$ where $n$, the leading coefficient, is: [ANS]\nand $r$, the exponent of $a$, is: [ANS]\nand $s$, the exponent of $b$, is: [ANS]\nand finally $t$, the exponent of $c$, is: [ANS]\n[NOTE: Your answers cannot be algebraic expressions.]", "answer_v3": [ "72", "18", "14", "15" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Algebra_0095", "subject": "Algebra", "topic": "Properties of exponents, rational exponents and radicals", "subtopic": "Properties of exponents", "level": "2", "keywords": [ "exponent", "negative", "fraction", "simplify" ], "problem_v1": "Simplify the following expression, and write your answer using only positive exponents.\n$\\frac{t^{-5}}{\\left(t^{8}\\right)^{8}}=$ [ANS]", "answer_v1": [ "1/(t^69)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Simplify the following expression, and write your answer using only positive exponents.\n$\\frac{x^{-3}}{\\left(x^{12}\\right)^{5}}=$ [ANS]", "answer_v2": [ "1/(x^63)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Simplify the following expression, and write your answer using only positive exponents.\n$\\frac{y^{-3}}{\\left(y^{9}\\right)^{6}}=$ [ANS]", "answer_v3": [ "1/(y^57)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0096", "subject": "Algebra", "topic": "Properties of exponents, rational exponents and radicals", "subtopic": "Properties of exponents", "level": "2", "keywords": [ "multiply", "exponent", "simplify" ], "problem_v1": "Use the properties of exponents to simplify the following\n${r^{16}}\\cdot{r^{13}}\\cdot{r^{13}}=$ [ANS]", "answer_v1": [ "r^42" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Use the properties of exponents to simplify the following\n${y^{3}}\\cdot{y^{19}}\\cdot{y^{4}}=$ [ANS]", "answer_v2": [ "y^26" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Use the properties of exponents to simplify the following\n${r^{7}}\\cdot{r^{13}}\\cdot{r^{7}}=$ [ANS]", "answer_v3": [ "r^27" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0097", "subject": "Algebra", "topic": "Properties of exponents, rational exponents and radicals", "subtopic": "Properties of exponents", "level": "2", "keywords": [ "exponent", "simplify", "zero" ], "problem_v1": "Use the properties of exponents to simplify the following\n$38B^0=$ [ANS]", "answer_v1": [ "38" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use the properties of exponents to simplify the following\n$6q^0=$ [ANS]", "answer_v2": [ "6" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use the properties of exponents to simplify the following\n$17B^0=$ [ANS]", "answer_v3": [ "17" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0098", "subject": "Algebra", "topic": "Properties of exponents, rational exponents and radicals", "subtopic": "Properties of exponents", "level": "2", "keywords": [ "algebra", "exponent" ], "problem_v1": "The expression x^{3}\\left(\\frac{1}{7} x^{4}\\right)(42x^{-10}) equals $c/x^e$ where the coefficient $c$ is [ANS], the exponent $e$ is [ANS].", "answer_v1": [ "6", "3" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The expression x^{2}\\left(\\frac{1}{1} x^{3}\\right)(7x^{-12}) equals $c/x^e$ where the coefficient $c$ is [ANS], the exponent $e$ is [ANS].", "answer_v2": [ "7", "7" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The expression x^{2}\\left(\\frac{1}{3} x^{3}\\right)(18x^{-9}) equals $c/x^e$ where the coefficient $c$ is [ANS], the exponent $e$ is [ANS].", "answer_v3": [ "6", "4" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0099", "subject": "Algebra", "topic": "Properties of exponents, rational exponents and radicals", "subtopic": "Properties of exponents", "level": "2", "keywords": [ "exponent", "powers" ], "problem_v1": "Find $x$ if \\frac{(5)^x (5)^{1}}{(5)^{3}}=(5)^{6} $x=$ [ANS]", "answer_v1": [ "8" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find $x$ if \\frac{(2.3)^x (2.3)^{6}}{(2.3)^{-2}}=(2.3)^{3} $x=$ [ANS]", "answer_v2": [ "-5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find $x$ if \\frac{(3.2)^x (3.2)^{2}}{(3.2)^{1}}=(3.2)^{4} $x=$ [ANS]", "answer_v3": [ "3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0100", "subject": "Algebra", "topic": "Properties of exponents, rational exponents and radicals", "subtopic": "Properties of exponents", "level": "2", "keywords": [ "real numbers", "algebraic expressions" ], "problem_v1": "Multiply using the product rule: $(-8x^{5} y^{6})(-5x^{5} y^{3})$=[ANS]", "answer_v1": [ "40*x^10*y^9" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Multiply using the product rule: $(-8 a^{7} b^{4})(-9 a^{2} b^{7})$=[ANS]", "answer_v2": [ "72*a^9*b^11" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Multiply using the product rule: $(-7 a^{5} b^{5})(-3 a^{3} b^{3})$=[ANS]", "answer_v3": [ "21*a^8*b^8" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0101", "subject": "Algebra", "topic": "Properties of exponents, rational exponents and radicals", "subtopic": "Properties of exponents", "level": "2", "keywords": [ "exponents" ], "problem_v1": "Rewrite the following using a single exponent.\n$\\begin{array}{cccc}\\hline & 32^2 y^{10}=\\Big([ANS] \\Big) & & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "2*y", "10" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Rewrite the following using a single exponent.\n$\\begin{array}{cccc}\\hline & 4^2 y^{4}=\\Big([ANS] \\Big) & & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "2*y", "4" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Rewrite the following using a single exponent.\n$\\begin{array}{cccc}\\hline & 8^2 y^{6}=\\Big([ANS] \\Big) & & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "2*y", "6" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0102", "subject": "Algebra", "topic": "Properties of exponents, rational exponents and radicals", "subtopic": "Properties of exponents", "level": "3", "keywords": [], "problem_v1": "Evaluate: $ \\frac{2^{6}}{2^{3}}=$ [ANS]\nNote: Your answer should be completely simplified. Unsimplified answers will not be accepted.", "answer_v1": [ "8" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate: $ \\frac{2^{5}}{2^{4}}=$ [ANS]\nNote: Your answer should be completely simplified. Unsimplified answers will not be accepted.", "answer_v2": [ "2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate: $ \\frac{2^{5}}{2^{3}}=$ [ANS]\nNote: Your answer should be completely simplified. Unsimplified answers will not be accepted.", "answer_v3": [ "4" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0103", "subject": "Algebra", "topic": "Properties of exponents, rational exponents and radicals", "subtopic": "Properties of exponents", "level": "3", "keywords": [ "factoring", "expansion" ], "problem_v1": "Rewrite the following as a perfect square.\n$64 k^{18}=\\big($ [ANS] $\\big)^{2}$", "answer_v1": [ "8*k^9" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Rewrite the following as a perfect square.\n$169 a^{6}=\\big($ [ANS] $\\big)^{2}$", "answer_v2": [ "13*a^3" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Rewrite the following as a perfect square.\n$81 c^{10}=\\big($ [ANS] $\\big)^{2}$", "answer_v3": [ "9*c^5" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0104", "subject": "Algebra", "topic": "Properties of exponents, rational exponents and radicals", "subtopic": "Properties of exponents", "level": "2", "keywords": [ "factoring", "expansion" ], "problem_v1": "Rewrite the following as a difference of cubes.\n$125 k^{12}-27x^{21}=\\big($ [ANS] $\\big)^{3}-\\big($ [ANS] $\\big)^3$", "answer_v1": [ "5*k^4", "3*x^7" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Rewrite the following as a difference of cubes.\n$8 a^{30}-27 z^{12}=\\big($ [ANS] $\\big)^{3}-\\big($ [ANS] $\\big)^3$", "answer_v2": [ "2*a^10", "3*z^4" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Rewrite the following as a difference of cubes.\n$27 c^{9}-64x^{15}=\\big($ [ANS] $\\big)^{3}-\\big($ [ANS] $\\big)^3$", "answer_v3": [ "3*c^3", "4*x^5" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0105", "subject": "Algebra", "topic": "Properties of exponents, rational exponents and radicals", "subtopic": "Properties of rational exponents and radicals", "level": "2", "keywords": [ "exponents" ], "problem_v1": "Simplify the expression as much as possible and leave it without radicals.\n$\\big(125 L^{3/4} P \\big)^{4/3} \\big(P \\big)^{-4/3}=$ [ANS]", "answer_v1": [ "625*L" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Simplify the expression as much as possible and leave it without radicals.\n$\\big(8 L^{3/4} P \\big)^{4/3} \\big(P \\big)^{-4/3}=$ [ANS]", "answer_v2": [ "16*L" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Simplify the expression as much as possible and leave it without radicals.\n$\\big(27 L^{3/4} P \\big)^{4/3} \\big(P \\big)^{-4/3}=$ [ANS]", "answer_v3": [ "81*L" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0106", "subject": "Algebra", "topic": "Properties of exponents, rational exponents and radicals", "subtopic": "Properties of rational exponents and radicals", "level": "2", "keywords": [ "algebra", "exponent" ], "problem_v1": "The expression \\left(\\frac{81}{36}\\right)^{-1/2} equals [ANS]/[ANS].", "answer_v1": [ "2", "3" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The expression \\left(\\frac{9}{64}\\right)^{-1/2} equals [ANS]/[ANS].", "answer_v2": [ "8", "3" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The expression \\left(\\frac{25}{36}\\right)^{-1/2} equals [ANS]/[ANS].", "answer_v3": [ "6", "5" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0107", "subject": "Algebra", "topic": "Properties of exponents, rational exponents and radicals", "subtopic": "Properties of rational exponents and radicals", "level": "2", "keywords": [ "algebra", "powers" ], "problem_v1": "The expression \\sqrt[5]{x^5 y^4}=x^ry^s where $x$ and $y$ are non-negative real numbers. r, the exponent of x, is: [ANS]\ns, the exponent of y, is: [ANS]", "answer_v1": [ "1", "0.8" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The expression \\sqrt[2]{x^2 y^5}=x^ry^s where $x$ and $y$ are non-negative real numbers. r, the exponent of x, is: [ANS]\ns, the exponent of y, is: [ANS]", "answer_v2": [ "1", "2.5" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The expression \\sqrt[3]{x^3 y^4}=x^ry^s where $x$ and $y$ are non-negative real numbers. r, the exponent of x, is: [ANS]\ns, the exponent of y, is: [ANS]", "answer_v3": [ "1", "1.33333333333333" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0108", "subject": "Algebra", "topic": "Properties of exponents, rational exponents and radicals", "subtopic": "Properties of rational exponents and radicals", "level": "2", "keywords": [ "algebra", "powers" ], "problem_v1": "The expression \\sqrt[11]{v^{77}} equals [ANS]", "answer_v1": [ "v^7" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "The expression \\sqrt[5]{v^{55}} equals [ANS]", "answer_v2": [ "v^11" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "The expression \\sqrt[7]{v^{49}} equals [ANS]", "answer_v3": [ "v^7" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0109", "subject": "Algebra", "topic": "Properties of exponents, rational exponents and radicals", "subtopic": "Properties of rational exponents and radicals", "level": "2", "keywords": [ "exponents" ], "problem_v1": "Simplify the following expression as much as possible. Assume that all variables are positive.\n$\\sqrt[4]{64x^{5}} \\, \\sqrt[4]{64x^{7}}$=[ANS]", "answer_v1": [ "8*x^3" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Simplify the following expression as much as possible. Assume that all variables are positive.\n$\\sqrt[4]{25x^{3}} \\, \\sqrt[4]{25x^{13}}$=[ANS]", "answer_v2": [ "5*x^4" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Simplify the following expression as much as possible. Assume that all variables are positive.\n$\\sqrt[4]{36x^{3}} \\, \\sqrt[4]{36x^{9}}$=[ANS]", "answer_v3": [ "6*x^3" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0110", "subject": "Algebra", "topic": "Properties of exponents, rational exponents and radicals", "subtopic": "Properties of rational exponents and radicals", "level": "3", "keywords": [], "problem_v1": "Simplify. Assume that all expressions under radicals represent nonnegative numbers.\n$\\left(\\sqrt{7 p-6}-3\\right)^2=$ [ANS]\nWrite your answer using radical notation if necessary. Help:", "answer_v1": [ "7*p+3-6*sqrt(7*p-6)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Simplify. Assume that all expressions under radicals represent nonnegative numbers.\n$\\left(\\sqrt{9x-11}-3\\right)^2=$ [ANS]\nWrite your answer using radical notation if necessary. Help:", "answer_v2": [ "9*x-2-6*sqrt(9*x-11)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Simplify. Assume that all expressions under radicals represent nonnegative numbers.\n$\\left(\\sqrt{6 z-7}-1\\right)^2=$ [ANS]\nWrite your answer using radical notation if necessary. Help:", "answer_v3": [ "6*z-6-2*sqrt(6*z-7)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0111", "subject": "Algebra", "topic": "Properties of exponents, rational exponents and radicals", "subtopic": "Properties of rational exponents and radicals", "level": "3", "keywords": [], "problem_v1": "Simplify completely. If\n \\sqrt[3]{24 p^{12} v^{23}}=A \\sqrt[3]{B} then $A=$ [ANS], $B=$ [ANS]. Hint: It is possible that $A$ and/or $B$ could equal $1$.", "answer_v1": [ "2*p^4*v^7", "3*v^2" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Simplify completely. If\n \\sqrt[3]{-250x^{12} r^{10}}=A \\sqrt[3]{B} then $A=$ [ANS], $B=$ [ANS]. Hint: It is possible that $A$ and/or $B$ could equal $1$.", "answer_v2": [ "-5*x^4*r^3", "2*r" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Simplify completely. If\n \\sqrt[3]{16 z^{15} v^{29}}=A \\sqrt[3]{B} then $A=$ [ANS], $B=$ [ANS]. Hint: It is possible that $A$ and/or $B$ could equal $1$.", "answer_v3": [ "2*z^5*v^9", "2*v^2" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0112", "subject": "Algebra", "topic": "Properties of exponents, rational exponents and radicals", "subtopic": "Properties of rational exponents and radicals", "level": "2", "keywords": [ "radical expression", "rational powers" ], "problem_v1": "Simplify and write the following using a rational exponent. If\n \\sqrt[5]{10^{7}}=10^m then $m=$ [ANS]", "answer_v1": [ "7/5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Simplify and write the following using a rational exponent. If\n \\sqrt[10]{7^{7}}=7^m then $m=$ [ANS]", "answer_v2": [ "7/10" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Simplify and write the following using a rational exponent. If\n \\sqrt[4]{9^{5}}=9^m then $m=$ [ANS]", "answer_v3": [ "5/4" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0113", "subject": "Algebra", "topic": "Cartesian coordinate system", "subtopic": "Plotting points", "level": "1", "keywords": [ "coordinates", "cartesian plane" ], "problem_v1": "The point ${\\left(5,2\\right)}$ is in Quadrant [ANS]. The point ${\\left(-4,-4\\right)}$ is in Quadrant [ANS]. The point ${\\left(-3,3\\right)}$ is in Quadrant [ANS]. The point ${\\left(10,-5\\right)}$ is in Quadrant [ANS].", "answer_v1": [ "I", "III", "II", "IV" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "I", "II", "III", "IV" ], [ "I", "II", "III", "IV" ], [ "I", "II", "III", "IV" ], [ "I", "II", "III", "IV" ] ], "problem_v2": "The point ${\\left(-9,9\\right)}$ is in Quadrant [ANS]. The point ${\\left(-7,-3\\right)}$ is in Quadrant [ANS]. The point ${\\left(9,-4\\right)}$ is in Quadrant [ANS]. The point ${\\left(4,9\\right)}$ is in Quadrant [ANS].", "answer_v2": [ "II", "III", "IV", "I" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "I", "II", "III", "IV" ], [ "I", "II", "III", "IV" ], [ "I", "II", "III", "IV" ], [ "I", "II", "III", "IV" ] ], "problem_v3": "The point ${\\left(-4,2\\right)}$ is in Quadrant [ANS]. The point ${\\left(-6,-3\\right)}$ is in Quadrant [ANS]. The point ${\\left(6,9\\right)}$ is in Quadrant [ANS]. The point ${\\left(8,-6\\right)}$ is in Quadrant [ANS].", "answer_v3": [ "II", "III", "I", "IV" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "I", "II", "III", "IV" ], [ "I", "II", "III", "IV" ], [ "I", "II", "III", "IV" ], [ "I", "II", "III", "IV" ] ] }, { "id": "Algebra_0114", "subject": "Algebra", "topic": "Cartesian coordinate system", "subtopic": "Plotting points", "level": "2", "keywords": [ "algebra", "coordinate", "Geometry", "Cartesian coordinates" ], "problem_v1": "Sketch the region given by the set $\\lbrace (x,y) | 3\\le x\\le 7, 2\\le y\\le 7 \\rbrace$ on a piece of paper. The area of the region is [ANS].", "answer_v1": [ "20" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Sketch the region given by the set $\\lbrace (x,y) |-5\\le x\\le 1,-5\\le y\\le-2 \\rbrace$ on a piece of paper. The area of the region is [ANS].", "answer_v2": [ "18" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Sketch the region given by the set $\\lbrace (x,y) |-2\\le x\\le 2,-3\\le y\\le 1 \\rbrace$ on a piece of paper. The area of the region is [ANS].", "answer_v3": [ "16" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0115", "subject": "Algebra", "topic": "Cartesian coordinate system", "subtopic": "Plotting points", "level": "2", "keywords": [ "Geometry", "Cartesian Coordinates" ], "problem_v1": "Match the set of points as indicated by the ordered pair with the correct quadrant the points lie in.\nPlace the letter of the part of the plane next to the corresponding description listed below: [ANS] 1. $(x,y)$, where $x<0, y>0$ [ANS] 2. $(x,y)$, where $x<0, y<0$ [ANS] 3. $(x,y)$, where $y=0$ [ANS] 4. $(x,y)$, where $x=0$\nA. Quadrant Two B. $x$-axis C. Quadrant Three D. $y$-axis", "answer_v1": [ "A", "C", "B", "D" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Match the set of points as indicated by the ordered pair with the correct quadrant the points lie in.\nPlace the letter of the part of the plane next to the corresponding description listed below: [ANS] 1. $(x,y)$, where $x<0, y<0$ [ANS] 2. $(x,y)$, where $x=0$ [ANS] 3. $(x,y)$, where $x<0, y>0$ [ANS] 4. $(x,y)$, where $x>0, y>0$\nA. Quadrant Two B. Quadrant One C. $y$-axis D. Quadrant Three", "answer_v2": [ "D", "C", "A", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Match the set of points as indicated by the ordered pair with the correct quadrant the points lie in.\nPlace the letter of the part of the plane next to the corresponding description listed below: [ANS] 1. $(x,y)$, where $x>0, y<0$ [ANS] 2. $(x,y)$, where $x>0, y>0$ [ANS] 3. $(x,y)$, where $x<0, y>0$ [ANS] 4. $(x,y)$, where $y=0$\nA. Quadrant Four B. Quadrant One C. $x$-axis D. Quadrant Two", "answer_v3": [ "A", "B", "D", "C" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Algebra_0116", "subject": "Algebra", "topic": "Cartesian coordinate system", "subtopic": "Plotting points", "level": "2", "keywords": [ "algebra", "coordinate", "points", "graph" ], "problem_v1": "For the point $P=\\left(-2,-4\\right)$, determine the points that are symmetric with respect to the $x$-axis, $y$-axis, and the origin. 1. [ANS] is symmetric to $P$ with respect to the $x$-axis. 2. [ANS] is symmetric to $P$ with respect to the $y$-axis. 3. [ANS] is symmetric to $P$ with respect to the origin. Note: Write your answer in the form $(a,b)$ where $a$ and $b$ are numbers.", "answer_v1": [ "(-2,-(-4))", "(-(-2),-4)", "(-(-2),4)" ], "answer_type_v1": [ "OL", "OL", "OL" ], "options_v1": [ [], [], [] ], "problem_v2": "For the point $P=\\left(-5,-2\\right)$, determine the points that are symmetric with respect to the $x$-axis, $y$-axis, and the origin. 1. [ANS] is symmetric to $P$ with respect to the $x$-axis. 2. [ANS] is symmetric to $P$ with respect to the $y$-axis. 3. [ANS] is symmetric to $P$ with respect to the origin. Note: Write your answer in the form $(a,b)$ where $a$ and $b$ are numbers.", "answer_v2": [ "(-5,-(-2))", "(-(-5),-2)", "(-(-5),2)" ], "answer_type_v2": [ "OL", "OL", "OL" ], "options_v2": [ [], [], [] ], "problem_v3": "For the point $P=\\left(-4,-3\\right)$, determine the points that are symmetric with respect to the $x$-axis, $y$-axis, and the origin. 1. [ANS] is symmetric to $P$ with respect to the $x$-axis. 2. [ANS] is symmetric to $P$ with respect to the $y$-axis. 3. [ANS] is symmetric to $P$ with respect to the origin. Note: Write your answer in the form $(a,b)$ where $a$ and $b$ are numbers.", "answer_v3": [ "(-4,-(-3))", "(-(-4),-3)", "(-(-4),3)" ], "answer_type_v3": [ "OL", "OL", "OL" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0117", "subject": "Algebra", "topic": "Cartesian coordinate system", "subtopic": "Plotting points", "level": "3", "keywords": [ "spheres" ], "problem_v1": "What center should a sphere of radius 4 have so that it sits on the xy plane with its bottom (lowest point in the z direction) just touching the origin? Answer: ([ANS], [ANS], [ANS]) What center should a sphere of radius 6 have so that it sits on top of the previous sphere (that is, higher in the z direction, touching the previous sphere at just one point)? Answer: ([ANS], [ANS], [ANS]) What are the two opposite corners of a box that tightly contains the previous two spheres (and whose edges are parallel to the axes)? First give the corner with all positive coordinates: ([ANS], [ANS], [ANS]) Then give the corner opposite that one: ([ANS], [ANS], [ANS]) What center should a sphere of radius 9 have so that it sits on top of the point $(-4,-4,1)$? Answer: ([ANS], [ANS], [ANS]) What center should a sphere of radius 13 have so that it hangs just below the previous sphere, touching it at just one point? Answer: ([ANS], [ANS], [ANS])", "answer_v1": [ "0", "0", "4", "0", "0", "14", "6", "6", "20", "-6", "-6", "0", "-4", "-4", "10", "-4", "-4", "-12" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "What center should a sphere of radius 2 have so that it sits on the xy plane with its bottom (lowest point in the z direction) just touching the origin? Answer: ([ANS], [ANS], [ANS]) What center should a sphere of radius 7 have so that it sits on top of the previous sphere (that is, higher in the z direction, touching the previous sphere at just one point)? Answer: ([ANS], [ANS], [ANS]) What are the two opposite corners of a box that tightly contains the previous two spheres (and whose edges are parallel to the axes)? First give the corner with all positive coordinates: ([ANS], [ANS], [ANS]) Then give the corner opposite that one: ([ANS], [ANS], [ANS]) What center should a sphere of radius 8 have so that it sits on top of the point $(9,-4,-7)$? Answer: ([ANS], [ANS], [ANS]) What center should a sphere of radius 12 have so that it hangs just below the previous sphere, touching it at just one point? Answer: ([ANS], [ANS], [ANS])", "answer_v2": [ "0", "0", "2", "0", "0", "11", "7", "7", "18", "-7", "-7", "0", "9", "-4", "1", "9", "-4", "-19" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "What center should a sphere of radius 2 have so that it sits on the xy plane with its bottom (lowest point in the z direction) just touching the origin? Answer: ([ANS], [ANS], [ANS]) What center should a sphere of radius 6 have so that it sits on top of the previous sphere (that is, higher in the z direction, touching the previous sphere at just one point)? Answer: ([ANS], [ANS], [ANS]) What are the two opposite corners of a box that tightly contains the previous two spheres (and whose edges are parallel to the axes)? First give the corner with all positive coordinates: ([ANS], [ANS], [ANS]) Then give the corner opposite that one: ([ANS], [ANS], [ANS]) What center should a sphere of radius 8 have so that it sits on top of the point $(-6,-3,6)$? Answer: ([ANS], [ANS], [ANS]) What center should a sphere of radius 12 have so that it hangs just below the previous sphere, touching it at just one point? Answer: ([ANS], [ANS], [ANS])", "answer_v3": [ "0", "0", "2", "0", "0", "10", "6", "6", "16", "-6", "-6", "0", "-6", "-3", "14", "-6", "-3", "-6" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Algebra_0118", "subject": "Algebra", "topic": "Cartesian coordinate system", "subtopic": "Midpoint and distance formulas", "level": "3", "keywords": [ "algebra", "distance" ], "problem_v1": "Find all $y$ such that the distance between the points $(5, 2)$ and $(3, y)$ is 22. $y=$ [ANS]\nNote: Enter your answer as a comma separated list of numbers. If there are no such $y$, enter none. none.", "answer_v1": [ "(2+sqrt(480), 2-sqrt(480))" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find all $y$ such that the distance between the points $(-9, 9)$ and $(-7, y)$ is 11. $y=$ [ANS]\nNote: Enter your answer as a comma separated list of numbers. If there are no such $y$, enter none. none.", "answer_v2": [ "(9+sqrt(117), 9-sqrt(117))" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find all $y$ such that the distance between the points $(-4, 2)$ and $(-5, y)$ is 17. $y=$ [ANS]\nNote: Enter your answer as a comma separated list of numbers. If there are no such $y$, enter none. none.", "answer_v3": [ "(2+sqrt(288), 2-sqrt(288))" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Algebra_0119", "subject": "Algebra", "topic": "Cartesian coordinate system", "subtopic": "Midpoint and distance formulas", "level": "2", "keywords": [ "coordinate geometry", "Manhattan distance" ], "problem_v1": "The Manhattan distance between the points $(x_1,y_1)$ and $(x_2, y_2)$ is defined as: D_M=|x_1-x_2|+|y_1-y_2|\nThe Manhattan distance between $(200, 95)$ and $(62, 172)$ is [ANS].", "answer_v1": [ "215" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The Manhattan distance between the points $(x_1,y_1)$ and $(x_2, y_2)$ is defined as: D_M=|x_1-x_2|+|y_1-y_2|\nThe Manhattan distance between $(-334,-290)$ and $(349,-140)$ is [ANS].", "answer_v2": [ "833" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The Manhattan distance between the points $(x_1,y_1)$ and $(x_2, y_2)$ is defined as: D_M=|x_1-x_2|+|y_1-y_2|\nThe Manhattan distance between $(-151,-180)$ and $(83, 42)$ is [ANS].", "answer_v3": [ "456" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0120", "subject": "Algebra", "topic": "Cartesian coordinate system", "subtopic": "Midpoint and distance formulas", "level": "2", "keywords": [], "problem_v1": "Plot the points $A=(1, 0)$, $B=(4, 3)$, and $C=(-5, 6)$. Notice that these points are vertices of a right triangle (the angle $A$ is 90 degrees). Find the distance between $A$ and $B$: [ANS]\nFind the distance between $A$ and $C$: [ANS]\nFind the area of the triangle $ABC$: [ANS]", "answer_v1": [ "4.24264068711928", "8.48528137423857", "18" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Plot the points $A=(-2, 2)$, $B=(-1, 4)$, and $C=(2, 0)$. Notice that these points are vertices of a right triangle (the angle $A$ is 90 degrees). Find the distance between $A$ and $B$: [ANS]\nFind the distance between $A$ and $C$: [ANS]\nFind the area of the triangle $ABC$: [ANS]", "answer_v2": [ "2.23606797749979", "4.47213595499958", "5" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Plot the points $A=(-1, 1)$, $B=(1, 4)$, and $C=(-7, 5)$. Notice that these points are vertices of a right triangle (the angle $A$ is 90 degrees). Find the distance between $A$ and $B$: [ANS]\nFind the distance between $A$ and $C$: [ANS]\nFind the area of the triangle $ABC$: [ANS]", "answer_v3": [ "3.60555127546399", "7.21110255092798", "13" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0121", "subject": "Algebra", "topic": "Cartesian coordinate system", "subtopic": "Midpoint and distance formulas", "level": "2", "keywords": [ "algebra", "coordinate geometry", "distance", "midpoint" ], "problem_v1": "The midpoint of $AB$ is at $(3, 1)$. If $A=(3,5)$, find $B$.\nB is:([ANS], [ANS])", "answer_v1": [ "3", "-3" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The midpoint of $AB$ is at $(-5, 5)$. If $A=(-7,-3)$, find $B$.\nB is:([ANS], [ANS])", "answer_v2": [ "-3", "13" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The midpoint of $AB$ is at $(-2, 1)$. If $A=(-5,1)$, find $B$.\nB is:([ANS], [ANS])", "answer_v3": [ "1", "1" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0122", "subject": "Algebra", "topic": "Cartesian coordinate system", "subtopic": "Midpoint and distance formulas", "level": "3", "keywords": [ "Geometry", "Cartesian" ], "problem_v1": "A convertible and a minivan leave a highway junction at the same time. The convertible travels west at 70 miles per hour and the minivan travels north at 60 miles per hour. Assuming the two vehicles do not deviate off course, how far apart are they after 3 hours? Distance Apart=[ANS]", "answer_v1": [ "276.586333718787" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A convertible and a minivan leave a highway junction at the same time. The convertible travels west at 40 miles per hour and the minivan travels north at 80 miles per hour. Assuming the two vehicles do not deviate off course, how far apart are they after 1.5 hours? Distance Apart=[ANS]", "answer_v2": [ "134.164078649987" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A convertible and a minivan leave a highway junction at the same time. The convertible travels west at 50 miles per hour and the minivan travels north at 60 miles per hour. Assuming the two vehicles do not deviate off course, how far apart are they after 1.5 hours? Distance Apart=[ANS]", "answer_v3": [ "117.1537451386" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0123", "subject": "Algebra", "topic": "Cartesian coordinate system", "subtopic": "Midpoint and distance formulas", "level": "2", "keywords": [ "Geometry", "Cartesian Coordinates", "Cartesian" ], "problem_v1": "The number of students enrolled at ASU in 1995 was 46208. In 2007, the number of students enrolled at ASU was 47930. Use the midpoint formula to estimate the number of students enrolled at ASU in 2001. Estimated number of students in 2001=[ANS]", "answer_v1": [ "47069" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The number of students enrolled at ASU in 1981 was 41495. In 2001, the number of students enrolled at ASU was 42829. Use the midpoint formula to estimate the number of students enrolled at ASU in 1991. Estimated number of students in 1991=[ANS]", "answer_v2": [ "42162" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The number of students enrolled at ASU in 1986 was 42786. In 2000, the number of students enrolled at ASU was 44334. Use the midpoint formula to estimate the number of students enrolled at ASU in 1993. Estimated number of students in 1993=[ANS]", "answer_v3": [ "43560" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0124", "subject": "Algebra", "topic": "Cartesian coordinate system", "subtopic": "Circles", "level": "3", "keywords": [ "algebra", "conic sections", "circle" ], "problem_v1": "Find the equation of the circle that has center $(3,1)$ and is tangent to the $y$-axis. Write it in the form (x-h)^2+(y-k)^2=r^2 and identify $h$, $k$, and $r$.\n$\\begin{array}{ccc}\\hline h &=& [ANS] \\\\ \\hline k &=& [ANS] \\\\ \\hline r &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "3", "1", "3" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the equation of the circle that has center $(-5,6)$ and is tangent to the $y$-axis. Write it in the form (x-h)^2+(y-k)^2=r^2 and identify $h$, $k$, and $r$.\n$\\begin{array}{ccc}\\hline h &=& [ANS] \\\\ \\hline k &=& [ANS] \\\\ \\hline r &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "-5", "6", "5" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the equation of the circle that has center $(-2,1)$ and is tangent to the $y$-axis. Write it in the form (x-h)^2+(y-k)^2=r^2 and identify $h$, $k$, and $r$.\n$\\begin{array}{ccc}\\hline h &=& [ANS] \\\\ \\hline k &=& [ANS] \\\\ \\hline r &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "-2", "1", "2" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0125", "subject": "Algebra", "topic": "Cartesian coordinate system", "subtopic": "Graphs of equations", "level": "2", "keywords": [ "algebra", "graph" ], "problem_v1": "For the graph of the equation $y=\\sqrt{x+10}$, answer the following questions: The $x$-intercepts have $x=$ [ANS]\nNote: If there is more than one answer enter them separated by commas. If there are none, enter none. The $y$-intercepts have $y=$ [ANS]\nNote: If there is more than one answer enter them separated by commas. If there are none, enter none. Is the graph symmetric with respect to the $x$-axis? Input yes or no here: [ANS]\nIs the graph symmetric with respect to the $y$-axis? Input yes or no here: [ANS]\nIs the graph symmetric with respect to the origin? Input yes or no here: [ANS]", "answer_v1": [ "-10", "3.16227766016838", "NO", "no", "no" ], "answer_type_v1": [ "NV", "NV", "TF", "TF", "TF" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "For the graph of the equation $y=\\sqrt{x+1}$, answer the following questions: The $x$-intercepts have $x=$ [ANS]\nNote: If there is more than one answer enter them separated by commas. If there are none, enter none. The $y$-intercepts have $y=$ [ANS]\nNote: If there is more than one answer enter them separated by commas. If there are none, enter none. Is the graph symmetric with respect to the $x$-axis? Input yes or no here: [ANS]\nIs the graph symmetric with respect to the $y$-axis? Input yes or no here: [ANS]\nIs the graph symmetric with respect to the origin? Input yes or no here: [ANS]", "answer_v2": [ "-1", "1", "NO", "no", "no" ], "answer_type_v2": [ "NV", "NV", "TF", "TF", "TF" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "For the graph of the equation $y=\\sqrt{x+4}$, answer the following questions: The $x$-intercepts have $x=$ [ANS]\nNote: If there is more than one answer enter them separated by commas. If there are none, enter none. The $y$-intercepts have $y=$ [ANS]\nNote: If there is more than one answer enter them separated by commas. If there are none, enter none. Is the graph symmetric with respect to the $x$-axis? Input yes or no here: [ANS]\nIs the graph symmetric with respect to the $y$-axis? Input yes or no here: [ANS]\nIs the graph symmetric with respect to the origin? Input yes or no here: [ANS]", "answer_v3": [ "-4", "2", "NO", "no", "no" ], "answer_type_v3": [ "NV", "NV", "TF", "TF", "TF" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Algebra_0126", "subject": "Algebra", "topic": "Cartesian coordinate system", "subtopic": "Graphs of equations", "level": "2", "keywords": [], "problem_v1": "Find the slope and $y$ intercept for each of the following lines.\nFor $3 y-12x=6 \\ \\ $, slope=[ANS] and y intercept=[ANS]. For $y=8x+3 \\ \\ $, slope=[ANS] and y intercept=[ANS]. For $3 y-24x=-15 \\ $, slope=[ANS] and y intercept=[ANS]. For $3 y+12x=6\\ $, slope=[ANS] and y intercept=[ANS]. For $y=5x+1 \\ $, slope=[ANS] and y intercept=[ANS].", "answer_v1": [ "4", "2", "8", "3", "8", "-5", "-4", "2", "5", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Find the slope and $y$ intercept for each of the following lines.\nFor $y=2x+5 \\ \\ $, slope=[ANS] and y intercept=[ANS]. For $y=7x-1 \\ \\ $, slope=[ANS] and y intercept=[ANS]. For $2 y-12x=4 \\ $, slope=[ANS] and y intercept=[ANS]. For $4 y+16x=20\\ $, slope=[ANS] and y intercept=[ANS]. For $2 y-14x=-6 \\ $, slope=[ANS] and y intercept=[ANS].", "answer_v2": [ "2", "5", "7", "-1", "6", "2", "-4", "5", "7", "-3" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Find the slope and $y$ intercept for each of the following lines.\nFor $4 y-12x=20 \\ \\ $, slope=[ANS] and y intercept=[ANS]. For $2 y+10x=4\\ \\ $, slope=[ANS] and y intercept=[ANS]. For $2 y-18x=4 \\ $, slope=[ANS] and y intercept=[ANS]. For $y=7x+1 \\ $, slope=[ANS] and y intercept=[ANS]. For $y=3x+1 \\ $, slope=[ANS] and y intercept=[ANS].", "answer_v3": [ "3", "5", "-5", "2", "9", "2", "7", "1", "3", "1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Algebra_0127", "subject": "Algebra", "topic": "Factoring", "subtopic": "Factoring: common factors", "level": "3", "keywords": [ "exponents", "factoring", "greatest common factor" ], "problem_v1": "Find the greatest common factor of the following three terms\n${8x^{13}y^{9}}$, ${-64x^{11}y^{14}}$, ${56x^{5}y^{18}}$ [ANS]", "answer_v1": [ "8*x^5*y^9" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the greatest common factor of the following three terms\n${5x^{20}y^{3}}$, ${-40x^{11}y^{9}}$, ${20x^{7}y^{10}}$ [ANS]", "answer_v2": [ "5*x^7*y^3" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the greatest common factor of the following three terms\n${6x^{12}y^{4}}$, ${-42x^{11}y^{11}}$, ${60x^{9}y^{13}}$ [ANS]", "answer_v3": [ "6*x^9*y^4" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0128", "subject": "Algebra", "topic": "Factoring", "subtopic": "Factoring: common factors", "level": "3", "keywords": [ "polynomial", "exponents", "factoring", "factor by grouping" ], "problem_v1": "Factor the given polynomial\n${t\\!\\left(t+7\\right)-8\\!\\left(t+7\\right)}=$ [ANS]\nIf the expression cannot be factored then answer with prime.", "answer_v1": [ "(t+7)*(t-8)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Factor the given polynomial\n${x\\!\\left(x-10\\right)+5\\!\\left(x-10\\right)}=$ [ANS]\nIf the expression cannot be factored then answer with prime.", "answer_v2": [ "(x-10)*(x+5)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Factor the given polynomial\n${y\\!\\left(y-7\\right)-6\\!\\left(y-7\\right)}=$ [ANS]\nIf the expression cannot be factored then answer with prime.", "answer_v3": [ "(y-7)*(y-6)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0129", "subject": "Algebra", "topic": "Factoring", "subtopic": "Factoring: common factors", "level": "3", "keywords": [ "algebra", "factoring" ], "problem_v1": "Factor the polynomial $(x-5)(x+4)^2-(x-5)^2(x+4)$. Your answer can be written as $A(x+B)(x+C)$ with integers $A$, $B$, $C$ where $A=$ [ANS], $B=$ [ANS], and $C=$ [ANS]", "answer_v1": [ "9", "-5", "4" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Factor the polynomial $(x-1)(x+6)^2-(x-1)^2(x+6)$. Your answer can be written as $A(x+B)(x+C)$ with integers $A$, $B$, $C$ where $A=$ [ANS], $B=$ [ANS], and $C=$ [ANS]", "answer_v2": [ "7", "-1", "6" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Factor the polynomial $(x-1)(x+4)^2-(x-1)^2(x+4)$. Your answer can be written as $A(x+B)(x+C)$ with integers $A$, $B$, $C$ where $A=$ [ANS], $B=$ [ANS], and $C=$ [ANS]", "answer_v3": [ "5", "-1", "4" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0130", "subject": "Algebra", "topic": "Factoring", "subtopic": "Factoring: common factors", "level": "2", "keywords": [], "problem_v1": "Here you are asked first the highest divisor to check. Answer that with a number. Then you are told that the number is prime. Answer that T (for true) or F (for false). To see if 121 is prime the we only have to check prime divisors up to [ANS]\n121 is prime: [ANS]\nTo see if 107 is prime the we only have to check prime divisors up to [ANS]\n107 is prime: [ANS]", "answer_v1": [ "11", "F", "7", "T" ], "answer_type_v1": [ "NV", "TF", "NV", "TF" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Here you are asked first the highest divisor to check. Answer that with a number. Then you are told that the number is prime. Answer that T (for true) or F (for false). To see if 87 is prime the we only have to check prime divisors up to [ANS]\n87 is prime: [ANS]\nTo see if 127 is prime the we only have to check prime divisors up to [ANS]\n127 is prime: [ANS]", "answer_v2": [ "7", "F", "11", "T" ], "answer_type_v2": [ "NV", "TF", "NV", "TF" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Here you are asked first the highest divisor to check. Answer that with a number. Then you are told that the number is prime. Answer that T (for true) or F (for false). To see if 117 is prime the we only have to check prime divisors up to [ANS]\n117 is prime: [ANS]\nTo see if 107 is prime the we only have to check prime divisors up to [ANS]\n107 is prime: [ANS]", "answer_v3": [ "7", "F", "7", "T" ], "answer_type_v3": [ "NV", "TF", "NV", "TF" ], "options_v3": [ [], [], [], [] ] }, { "id": "Algebra_0131", "subject": "Algebra", "topic": "Factoring", "subtopic": "Factoring: common factors", "level": "2", "keywords": [], "problem_v1": "Factor each monomial or number completely. list the factors separated by commas. (For example an answer might be 2,2,3,x,x). Remember that if a monomial or number is negative then one of the factors is-1, all other factors must be positive. (In such a case and answer might be-1,2,5,x,y,y.) The factors of 115 are [ANS]\nThe factors of-133 are [ANS]\nThe factors of-18 are [ANS]\nThe factors of 195 are [ANS]", "answer_v1": [ "(5, 23)", "(-1, 7, 19)", "(-1, 3, 3, 2)", "(3, 5, 13)" ], "answer_type_v1": [ "UOL", "UOL", "UOL", "UOL" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Factor each monomial or number completely. list the factors separated by commas. (For example an answer might be 2,2,3,x,x). Remember that if a monomial or number is negative then one of the factors is-1, all other factors must be positive. (In such a case and answer might be-1,2,5,x,y,y.) The factors of 87 are [ANS]\nThe factors of-85 are [ANS]\nThe factors of-54 are [ANS]\nThe factors of 102 are [ANS]", "answer_v2": [ "(3, 29)", "(-1, 5, 17)", "(-1, 3, 3, 3, 2)", "(2, 3, 17)" ], "answer_type_v2": [ "UOL", "UOL", "UOL", "UOL" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Factor each monomial or number completely. list the factors separated by commas. (For example an answer might be 2,2,3,x,x). Remember that if a monomial or number is negative then one of the factors is-1, all other factors must be positive. (In such a case and answer might be-1,2,5,x,y,y.) The factors of 69 are [ANS]\nThe factors of-95 are [ANS]\nThe factors of-117 are [ANS]\nThe factors of 255 are [ANS]", "answer_v3": [ "(3, 23)", "(-1, 5, 19)", "(-1, 3, 3, 13)", "(3, 5, 17)" ], "answer_type_v3": [ "UOL", "UOL", "UOL", "UOL" ], "options_v3": [ [], [], [], [] ] }, { "id": "Algebra_0132", "subject": "Algebra", "topic": "Factoring", "subtopic": "Factoring: common factors", "level": "3", "keywords": [ "Equations" ], "problem_v1": "Rewrite the expression $8 r (s-3)-24 (s-3)$ by taking out the greatest common factor.\n$\\big($ [ANS] $\\big) \\big($ [ANS] $\\big)$", "answer_v1": [ "8*(s-3)", "r-3" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Rewrite the expression $4 r (s-3)-8 (s-3)$ by taking out the greatest common factor.\n$\\big($ [ANS] $\\big) \\big($ [ANS] $\\big)$", "answer_v2": [ "4*(s-3)", "r-2" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Rewrite the expression $5 r (s-3)-10 (s-3)$ by taking out the greatest common factor.\n$\\big($ [ANS] $\\big) \\big($ [ANS] $\\big)$", "answer_v3": [ "5*(s-3)", "r-2" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0133", "subject": "Algebra", "topic": "Factoring", "subtopic": "Factoring: common factors", "level": "3", "keywords": [ "quadratic functions" ], "problem_v1": "(a) Factor the quadratic function $y=x (18x-15)-6 (6x-5)$. $y$=[ANS]\n(b) Find the zeros of $y=x (18x-15)-6 (6x-5)$. If there is more than one correct answer, enter your answers as a comma separated list. If there are no solutions, enter NONE. $x=$ [ANS]", "answer_v1": [ "(3*x-6)*(6*x-5)", "(2, 0.833333)" ], "answer_type_v1": [ "EX", "UOL" ], "options_v1": [ [], [] ], "problem_v2": "(a) Factor the quadratic function $y=x (4x-14)-4 (2x-7)$. $y$=[ANS]\n(b) Find the zeros of $y=x (4x-14)-4 (2x-7)$. If there is more than one correct answer, enter your answers as a comma separated list. If there are no solutions, enter NONE. $x=$ [ANS]", "answer_v2": [ "(2*x-4)*(2*x-7)", "(2, 3.5)" ], "answer_type_v2": [ "EX", "UOL" ], "options_v2": [ [], [] ], "problem_v3": "(a) Factor the quadratic function $y=x (6x-10)-4 (3x-5)$. $y$=[ANS]\n(b) Find the zeros of $y=x (6x-10)-4 (3x-5)$. If there is more than one correct answer, enter your answers as a comma separated list. If there are no solutions, enter NONE. $x=$ [ANS]", "answer_v3": [ "(2*x-4)*(3*x-5)", "(2, 1.66667)" ], "answer_type_v3": [ "EX", "UOL" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0134", "subject": "Algebra", "topic": "Factoring", "subtopic": "Factoring by grouping", "level": "2", "keywords": [ "factoring", "factor", "factorization", "substitution" ], "problem_v1": "Factor the expression and simplify your answer as much as possible:\n$6 s^2+s-1=\\big($ [ANS] $\\big) \\big($ [ANS] $\\big)$", "answer_v1": [ "2*s+1", "3*s-1" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Factor the expression and simplify your answer as much as possible:\n$2 b^2-3 b-9=\\big($ [ANS] $\\big) \\big($ [ANS] $\\big)$", "answer_v2": [ "2*b+3", "b-3" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Factor the expression and simplify your answer as much as possible:\n$4x^2-5x-6=\\big($ [ANS] $\\big) \\big($ [ANS] $\\big)$", "answer_v3": [ "4*x+3", "x-2" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0135", "subject": "Algebra", "topic": "Factoring", "subtopic": "Factoring trinomials", "level": "2", "keywords": [ "Equations" ], "problem_v1": "Factor the expression $n^2-n-42$. Simplify your answer as much as possible, but do not combine like factors. [ANS]", "answer_v1": [ "(n+6)*(n-7)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Factor the expression $n^2-n-12$. Simplify your answer as much as possible, but do not combine like factors. [ANS]", "answer_v2": [ "(n+3)*(n-4)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Factor the expression $n^2-n-20$. Simplify your answer as much as possible, but do not combine like factors. [ANS]", "answer_v3": [ "(n+4)*(n-5)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0136", "subject": "Algebra", "topic": "Factoring", "subtopic": "Factoring trinomials", "level": "2", "keywords": [ "algebra", "factoring", "quadratic" ], "problem_v1": "Factor out the greatest common factor first and place it in front. Then factoring the remaining expression as much as possible, and type your result in the second box:\n$5x^2+25x+30=$ [ANS]", "answer_v1": [ "5(x+2)*(x+3)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Factor out the greatest common factor first and place it in front. Then factoring the remaining expression as much as possible, and type your result in the second box:\n$2x^2+4x-96=$ [ANS]", "answer_v2": [ "2(x+8)*(x+-6)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Factor out the greatest common factor first and place it in front. Then factoring the remaining expression as much as possible, and type your result in the second box:\n$3x^2-3x-36=$ [ANS]", "answer_v3": [ "3(x+3)*(x+-4)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0137", "subject": "Algebra", "topic": "Factoring", "subtopic": "Factoring: special forms", "level": "2", "keywords": [ "factoring" ], "problem_v1": "Factor the difference of squares: $36x^{2}-25y^{2}=$ [ANS]", "answer_v1": [ "(6*x-5*y)*(6*x+5*y)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Factor the difference of squares: $16x^{2}-49y^{2}=$ [ANS]", "answer_v2": [ "(4*x-7*y)*(4*x+7*y)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Factor the difference of squares: $16x^{2}-25y^{2}=$ [ANS]", "answer_v3": [ "(4*x-5*y)*(4*x+5*y)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0138", "subject": "Algebra", "topic": "Factoring", "subtopic": "Factoring: special forms", "level": "2", "keywords": [ "factoring", "complete the square" ], "problem_v1": "List all values of $c$ that makes each trinomial a perfect square trinomial. Separate multiple answers by commas.\n(a) $x^2+8x+c$: [ANS]\n(b) $x^2+c x+25$: [ANS]", "answer_v1": [ "16", "(-10, 10)" ], "answer_type_v1": [ "NV", "UOL" ], "options_v1": [ [], [] ], "problem_v2": "List all values of $c$ that makes each trinomial a perfect square trinomial. Separate multiple answers by commas.\n(a) $x^2+2x+c$: [ANS]\n(b) $x^2+c x+9$: [ANS]", "answer_v2": [ "1", "(-6, 6)" ], "answer_type_v2": [ "NV", "UOL" ], "options_v2": [ [], [] ], "problem_v3": "List all values of $c$ that makes each trinomial a perfect square trinomial. Separate multiple answers by commas.\n(a) $x^2+4x+c$: [ANS]\n(b) $x^2+c x+16$: [ANS]", "answer_v3": [ "4", "(-8, 8)" ], "answer_type_v3": [ "NV", "UOL" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0139", "subject": "Algebra", "topic": "Factoring", "subtopic": "Factoring polynomials: general", "level": "3", "keywords": [ "quadratic functions" ], "problem_v1": "Write the expression $25 t^2+80 t+64$ in factored form $k (at+b)(ct+d)$.\n$25 t^2+80 t+64$=[ANS]", "answer_v1": [ "(5*t+8)*(5*t+8)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Write the expression $9 t^2+60 t+100$ in factored form $k (at+b)(ct+d)$.\n$9 t^2+60 t+100$=[ANS]", "answer_v2": [ "(3*t+10)*(3*t+10)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Write the expression $9 t^2+54 t+81$ in factored form $k (at+b)(ct+d)$.\n$9 t^2+54 t+81$=[ANS]", "answer_v3": [ "(3*t+9)*(3*t+9)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0141", "subject": "Algebra", "topic": "Functions", "subtopic": "Definition, concept", "level": "2", "keywords": [ "Mod", "Modular" ], "problem_v1": "Determine whether f is a function from $\\mathbb{Z}$ to $\\mathbb{R}$. Enter \"Y\" for yes and \"N\" for no. [ANS] 1. $f(n)=1/(n^2+6)$ [ANS] 2. $f(n)=\\frac{1}{n^2-25}$ [ANS] 3. $f(n)=\\sqrt{n^2+6}$ [ANS] 4. $f(n)=\\pm n$", "answer_v1": [ "Y", "N", "Y", "N" ], "answer_type_v1": [ "TF", "TF", "TF", "TF" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Determine whether f is a function from $\\mathbb{Z}$ to $\\mathbb{R}$. Enter \"Y\" for yes and \"N\" for no. [ANS] 1. $f(n)=\\pm n$ [ANS] 2. $f(n)=1/(n^2+9)$ [ANS] 3. $f(n)=\\sqrt{n^2+2}$ [ANS] 4. $f(n)=\\frac{1}{n^2-4}$", "answer_v2": [ "N", "Y", "Y", "N" ], "answer_type_v2": [ "TF", "TF", "TF", "TF" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Determine whether f is a function from $\\mathbb{Z}$ to $\\mathbb{R}$. Enter \"Y\" for yes and \"N\" for no. [ANS] 1. $f(n)=1/(n^2+6)$ [ANS] 2. $f(n)=\\pm n$ [ANS] 3. $f(n)=\\frac{1}{n^2-9}$ [ANS] 4. $f(n)=\\sqrt{n^2+3}$", "answer_v3": [ "Y", "N", "N", "Y" ], "answer_type_v3": [ "TF", "TF", "TF", "TF" ], "options_v3": [ [], [], [], [] ] }, { "id": "Algebra_0142", "subject": "Algebra", "topic": "Functions", "subtopic": "Definition, concept", "level": "1", "keywords": [ "Functions", "Polynomial" ], "problem_v1": "Use the table below to answer the following questions. If there is more than one answer to a question, enter as a comma-separated list of values. Enter none if there is no answer.\n$\\begin{array}{ccccccccc}\\hline {\\large{x}} &-3 &-2 &-1 & 0 & 1 & 2 & 3 & 4 \\\\ \\hline {\\large{y}} & 1 & 4 & 2 &-2 & 4 & 3 & 0 & 7 \\\\ \\hline \\end{array}$\n$\\begin{array}{cc}\\hline\n(a) For what values of \\small{x \\;\\mbox{is}\\; y=4}? & [ANS] \\\\ \\hline (b) For what values of \\small{x \\;\\mbox{is}\\; y=5}? & [ANS] \\\\ \\hline (c) For what values of \\small{y \\;\\mbox{is}\\; x=1}? & [ANS] \\\\ \\hline (d) For what values of \\small{x \\;\\mbox{is}\\; y \\le 0}? & [ANS] \\\\ \\hline (e) What is the minimum value of \\small{y}? & [ANS] \\\\ \\hline At what value of \\small{x}does it occur? & [ANS] \\\\ \\hline (f) What is the maximum value of \\small{y}? & [ANS] \\\\ \\hline At what value of \\small{x}does it occur? & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "(-2, 1)", "none", "4", "(0, 3)", "-2", "0", "7", "4" ], "answer_type_v1": [ "UOL", "OE", "NV", "UOL", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Use the table below to answer the following questions. If there is more than one answer to a question, enter as a comma-separated list of values. Enter none if there is no answer.\n$\\begin{array}{ccccccccc}\\hline {\\large{x}} &-2 &-1 & 0 & 1 & 2 & 3 & 4 & 5 \\\\ \\hline {\\large{y}} &-4 & 1 &-2 & 4 & 1 &-1 & 3 &-5 \\\\ \\hline \\end{array}$\n$\\begin{array}{cc}\\hline\n(a) For what values of \\small{x \\;\\mbox{is}\\; y=1}? & [ANS] \\\\ \\hline (b) For what values of \\small{x \\;\\mbox{is}\\; y=5}? & [ANS] \\\\ \\hline (c) For what values of \\small{y \\;\\mbox{is}\\; x=2}? & [ANS] \\\\ \\hline (d) For what values of \\small{x \\;\\mbox{is}\\; y \\le 0}? & [ANS] \\\\ \\hline (e) What is the minimum value of \\small{y}? & [ANS] \\\\ \\hline At what value of \\small{x}does it occur? & [ANS] \\\\ \\hline (f) What is the maximum value of \\small{y}? & [ANS] \\\\ \\hline At what value of \\small{x}does it occur? & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "(-1, 2)", "none", "1", "(-2, 0, 3, 5)", "-5", "5", "4", "1" ], "answer_type_v2": [ "UOL", "OE", "NV", "UOL", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Use the table below to answer the following questions. If there is more than one answer to a question, enter as a comma-separated list of values. Enter none if there is no answer.\n$\\begin{array}{ccccccccc}\\hline {\\large{x}} &-3 &-2 &-1 & 0 & 1 & 2 & 3 & 4 \\\\ \\hline {\\large{y}} &-3 & 2 & 0 & 1 & 2 & 3 & 4 &-1 \\\\ \\hline \\end{array}$\n$\\begin{array}{cc}\\hline\n(a) For what values of \\small{x \\;\\mbox{is}\\; y=2}? & [ANS] \\\\ \\hline (b) For what values of \\small{x \\;\\mbox{is}\\; y=5}? & [ANS] \\\\ \\hline (c) For what values of \\small{y \\;\\mbox{is}\\; x=1}? & [ANS] \\\\ \\hline (d) For what values of \\small{x \\;\\mbox{is}\\; y \\le 0}? & [ANS] \\\\ \\hline (e) What is the minimum value of \\small{y}? & [ANS] \\\\ \\hline At what value of \\small{x}does it occur? & [ANS] \\\\ \\hline (f) What is the maximum value of \\small{y}? & [ANS] \\\\ \\hline At what value of \\small{x}does it occur? & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "(-2, 1)", "none", "2", "(-3, -1, 4)", "-3", "-3", "4", "3" ], "answer_type_v3": [ "UOL", "OE", "NV", "UOL", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Algebra_0143", "subject": "Algebra", "topic": "Functions", "subtopic": "Definition, concept", "level": "2", "keywords": [ "algebra", "functions" ], "problem_v1": "If $f(x)=4x^2+x+2$, find the following:\n(a) $f(3)=$ [ANS]\n(b) $f(-3)=$ [ANS]\n(c) $f(-2)=$ [ANS]", "answer_v1": [ "41", "35", "16" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "If $f(x)=x^2+6x-5$, find the following:\n(a) $f(-2)=$ [ANS]\n(b) $f(6)=$ [ANS]\n(c) $f(-4)=$ [ANS]", "answer_v2": [ "-13", "67", "-13" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "If $f(x)=2x^2+2x-3$, find the following:\n(a) $f(1)=$ [ANS]\n(b) $f(-4)=$ [ANS]\n(c) $f(-2)=$ [ANS]", "answer_v3": [ "1", "21", "1" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0144", "subject": "Algebra", "topic": "Functions", "subtopic": "Function notation", "level": "3", "keywords": [ "algebra", "function" ], "problem_v1": "Let $f(x)=5x^{2}+4x+4$ and $ g(h)=\\frac {f(4+h)-f(4)}{h}$. Evaluate the following:\n$\\begin{array}{cccc}\\hline 1. & g(1) &=& [ANS] \\\\ \\hline 2. & g(0.1) &=& [ANS] \\\\ \\hline 3. & g(0.01) &=& [ANS] \\\\ \\hline 4. & g(0.001) &=& [ANS] \\\\ \\hline \\end{array}$\nNotice that the values that you entered are getting closer and closer to a number we'll call $L$. Find $L$. $L=$ [ANS]\nThis number is called the limit of $g(h)$ as $h$ approaches $0$. It is also called the derivative of $f(x)$ at the point when $x=4$. We will see more of this when we get to the definition of derivative.", "answer_v1": [ "49", "44.5", "44.05", "44.005", "44" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Let $f(x)=2x^{2}+5x+2$ and $ g(h)=\\frac {f(3+h)-f(3)}{h}$. Evaluate the following:\n$\\begin{array}{cccc}\\hline 1. & g(1) &=& [ANS] \\\\ \\hline 2. & g(0.1) &=& [ANS] \\\\ \\hline 3. & g(0.01) &=& [ANS] \\\\ \\hline 4. & g(0.001) &=& [ANS] \\\\ \\hline \\end{array}$\nNotice that the values that you entered are getting closer and closer to a number we'll call $L$. Find $L$. $L=$ [ANS]\nThis number is called the limit of $g(h)$ as $h$ approaches $0$. It is also called the derivative of $f(x)$ at the point when $x=3$. We will see more of this when we get to the definition of derivative.", "answer_v2": [ "19", "17.2", "17.02", "17.002", "17" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Let $f(x)=3x^{2}+4x+3$ and $ g(h)=\\frac {f(3+h)-f(3)}{h}$. Evaluate the following:\n$\\begin{array}{cccc}\\hline 1. & g(1) &=& [ANS] \\\\ \\hline 2. & g(0.1) &=& [ANS] \\\\ \\hline 3. & g(0.01) &=& [ANS] \\\\ \\hline 4. & g(0.001) &=& [ANS] \\\\ \\hline \\end{array}$\nNotice that the values that you entered are getting closer and closer to a number we'll call $L$. Find $L$. $L=$ [ANS]\nThis number is called the limit of $g(h)$ as $h$ approaches $0$. It is also called the derivative of $f(x)$ at the point when $x=3$. We will see more of this when we get to the definition of derivative.", "answer_v3": [ "25", "22.3", "22.03", "22.003", "22" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Algebra_0145", "subject": "Algebra", "topic": "Functions", "subtopic": "Function notation", "level": "2", "keywords": [ "evaluate", "function" ], "problem_v1": "Consider the function $H$ defined by $H(x)={\\sqrt{x}}$. Evaluate the following:\n$H(49)=$ [ANS]\n$H\\left({{\\textstyle\\frac{49}{64}}}\\right)=$ [ANS]", "answer_v1": [ "7", "7/8" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Consider the function $f$ defined by $f(x)={\\sqrt{x}}$. Evaluate the following:\n$f(100)=$ [ANS]\n$f\\left({{\\textstyle\\frac{4}{25}}}\\right)=$ [ANS]", "answer_v2": [ "10", "2/5" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Consider the function $h$ defined by $h(x)={\\sqrt{x}}$. Evaluate the following:\n$h(49)=$ [ANS]\n$h\\left({{\\textstyle\\frac{9}{25}}}\\right)=$ [ANS]", "answer_v3": [ "7", "3/5" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0146", "subject": "Algebra", "topic": "Functions", "subtopic": "Function notation", "level": "2", "keywords": [ "function", "evaluate" ], "problem_v1": "Evaluate the function $H$ at the given values.\n$H(r)={-r^{2}+3}$\n$H(4)=$ [ANS]\n$H(-4)=$ [ANS]\n$H(0)=$ [ANS]", "answer_v1": [ "-13", "-13", "3" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Evaluate the function $f$ at the given values.\n$f(t)={-t^{2}-7}$\n$f(2)=$ [ANS]\n$f(-1)=$ [ANS]\n$f(0)=$ [ANS]", "answer_v2": [ "-11", "-8", "-7" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Evaluate the function $h$ at the given values.\n$h(r)={-r^{2}-5}$\n$h(3)=$ [ANS]\n$h(-4)=$ [ANS]\n$h(0)=$ [ANS]", "answer_v3": [ "-14", "-21", "-5" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0147", "subject": "Algebra", "topic": "Functions", "subtopic": "Function notation", "level": "2", "keywords": [ "domain", "range", "function", "ordered pair" ], "problem_v1": "Fill in the blanks.\nIf $H(6)=7$, then the point [ANS] is on the graph of $H$. If ${\\left(8,6\\right)}$ is on the graph of $H$, then $H(8)=$ [ANS].", "answer_v1": [ "(6,7)", "6" ], "answer_type_v1": [ "OL", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Fill in the blanks.\nIf $f(10)=6$, then the point [ANS] is on the graph of $f$. If ${\\left(4,9\\right)}$ is on the graph of $f$, then $f(4)=$ [ANS].", "answer_v2": [ "(10,6)", "9" ], "answer_type_v2": [ "OL", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Fill in the blanks.\nIf $h(7)=5$, then the point [ANS] is on the graph of $h$. If ${\\left(6,3\\right)}$ is on the graph of $h$, then $h(6)=$ [ANS].", "answer_v3": [ "(7,5)", "3" ], "answer_type_v3": [ "OL", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0148", "subject": "Algebra", "topic": "Functions", "subtopic": "Function notation", "level": "1", "keywords": [ "functions", "definition of function", "function notation" ], "problem_v1": "Write the relationship below using function notation (i.e. $y$ is a function of $x$ is written in function notation as $y=f(x)$). Average score, $a$, is a function of number of students in the class, $n$. [ANS]=$f$ ([ANS]) (fill in the two blanks in the expression with the appropriate variables)", "answer_v1": [ "a", "n" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Write the relationship below using function notation (i.e. $y$ is a function of $x$ is written in function notation as $y=f(x)$). Weight, $w$, is a function of caloric intake, $c$. [ANS]=$f$ ([ANS]) (fill in the two blanks in the expression with the appropriate variables)", "answer_v2": [ "w", "c" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Write the relationship below using function notation (i.e. $y$ is a function of $x$ is written in function notation as $y=f(x)$). Volume, $v$, is a function of height, $h$. [ANS]=$f$ ([ANS]) (fill in the two blanks in the expression with the appropriate variables)", "answer_v3": [ "v", "h" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0149", "subject": "Algebra", "topic": "Functions", "subtopic": "Function notation", "level": "2", "keywords": [ "functions", "definition of function", "function notation" ], "problem_v1": "Consider the function $f(x)=\\frac{18}{5+x^2}$. Complete the table of values for $f(x)$ below when $x=0, 1, 2, 3$. Give exact values. $\\begin{array}{ccccc}\\hline x & 0 & 1 & 2 & 3 \\\\ \\hline f(x) & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "18/5", "18/6", "18/9", "18/14" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Consider the function $f(x)=\\frac{8}{7+x^2}$. Complete the table of values for $f(x)$ below when $x=0, 1, 2, 3$. Give exact values. $\\begin{array}{ccccc}\\hline x & 0 & 1 & 2 & 3 \\\\ \\hline f(x) & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "8/7", "8/8", "8/11", "8/16" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Consider the function $f(x)=\\frac{12}{5+x^2}$. Complete the table of values for $f(x)$ below when $x=0, 1, 2, 3$. Give exact values. $\\begin{array}{ccccc}\\hline x & 0 & 1 & 2 & 3 \\\\ \\hline f(x) & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "12/5", "12/6", "12/9", "12/14" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Algebra_0150", "subject": "Algebra", "topic": "Functions", "subtopic": "Function notation", "level": "2", "keywords": [ "functions", "evaluating", "solving", "input", "output" ], "problem_v1": "If $g(t)=\\frac{1}{t+8}-6$\n(a) Find $g(0)$. $g(0)=$ [ANS]\n(b) Exactly solve $g(t)=0$. If there is more than one solution, enter all solutions as a comma separated list of (exact) values. $t=$ [ANS]", "answer_v1": [ "-47/8", "-47/6" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "If $g(t)=\\frac{1}{t+2}-9$\n(a) Find $g(0)$. $g(0)=$ [ANS]\n(b) Exactly solve $g(t)=0$. If there is more than one solution, enter all solutions as a comma separated list of (exact) values. $t=$ [ANS]", "answer_v2": [ "-17/2", "-17/9" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "If $g(t)=\\frac{1}{t+4}-6$\n(a) Find $g(0)$. $g(0)=$ [ANS]\n(b) Exactly solve $g(t)=0$. If there is more than one solution, enter all solutions as a comma separated list of (exact) values. $t=$ [ANS]", "answer_v3": [ "-23/4", "-23/6" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0151", "subject": "Algebra", "topic": "Functions", "subtopic": "Function notation", "level": "1", "keywords": [ "algebra", "function", "definition", "Functions", "Evaluation" ], "problem_v1": "Express the rule \"Subtract 20, then square\" as the function\n$f(x)=$ [ANS].", "answer_v1": [ "(x-20)*(x-20)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Express the rule \"Subtract 3, then square\" as the function\n$f(x)=$ [ANS].", "answer_v2": [ "(x-3)*(x-3)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Express the rule \"Subtract 9, then square\" as the function\n$f(x)=$ [ANS].", "answer_v3": [ "(x-9)*(x-9)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0152", "subject": "Algebra", "topic": "Functions", "subtopic": "Function notation", "level": "2", "keywords": [ "Functions", "Piecewise", "Domain" ], "problem_v1": "Find $\\small{f(12), f(-7), f(\\pi), \\;\\mbox{and}\\; f(-7.1)}$ for:\n\\small{f(x)=\\begin{cases}{\\sqrt{x+7}}&\\text{if}\\ x > 7\\cr {2}&\\text{if}\\ x \\le 7\\end{cases}} You may keep radicals in any answers where appropriate. Use pi to represent $\\small{\\pi}$.\n$\\begin{array}{ccc}\\hline \\small{f(12)} &=& [ANS] \\\\ \\hline \\small{f(-7)} &=& [ANS] \\\\ \\hline \\small{f(\\pi)} &=& [ANS] \\\\ \\hline \\small{f(-7.1)} &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "4.3589", "2", "2", "2" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Find $\\small{f(4), f(-1), f(\\pi), \\;\\mbox{and}\\; f(-1.1)}$ for:\n\\small{f(x)=\\begin{cases}{\\sqrt{x+1}}&\\text{if}\\ x > 1\\cr {8}&\\text{if}\\ x \\le 1\\end{cases}} You may keep radicals in any answers where appropriate. Use pi to represent $\\small{\\pi}$.\n$\\begin{array}{ccc}\\hline \\small{f(4)} &=& [ANS] \\\\ \\hline \\small{f(-1)} &=& [ANS] \\\\ \\hline \\small{f(\\pi)} &=& [ANS] \\\\ \\hline \\small{f(-1.1)} &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "2.23607", "8", "sqrt(pi+1)", "8" ], "answer_type_v2": [ "NV", "NV", "EX", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Find $\\small{f(7), f(-3), f(\\pi), \\;\\mbox{and}\\; f(-3.1)}$ for:\n\\small{f(x)=\\begin{cases}{\\sqrt{x+3}}&\\text{if}\\ x > 3\\cr {2}&\\text{if}\\ x \\le 3\\end{cases}} You may keep radicals in any answers where appropriate. Use pi to represent $\\small{\\pi}$.\n$\\begin{array}{ccc}\\hline \\small{f(7)} &=& [ANS] \\\\ \\hline \\small{f(-3)} &=& [ANS] \\\\ \\hline \\small{f(\\pi)} &=& [ANS] \\\\ \\hline \\small{f(-3.1)} &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "3.16228", "2", "sqrt(pi+3)", "2" ], "answer_type_v3": [ "NV", "NV", "EX", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Algebra_0153", "subject": "Algebra", "topic": "Functions", "subtopic": "Function notation", "level": "2", "keywords": [ "algebra" ], "problem_v1": "For the next few problems you need to understand what it means to evaluate a function. You simply replace the variable with the number at which you evaluate the function. For example, the answer to the first question below is $54$ since 54=8*6+6. Let the function $f$ be defined by f(x)=8x+6. Then $f(6)=$ [ANS] and $f(7)=$ [ANS]", "answer_v1": [ "54", "62" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "For the next few problems you need to understand what it means to evaluate a function. You simply replace the variable with the number at which you evaluate the function. For example, the answer to the first question below is $15$ since 15=2*3+9. Let the function $f$ be defined by f(x)=2x+9. Then $f(3)=$ [ANS] and $f(4)=$ [ANS]", "answer_v2": [ "15", "17" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "For the next few problems you need to understand what it means to evaluate a function. You simply replace the variable with the number at which you evaluate the function. For example, the answer to the first question below is $22$ since 22=4*4+6. Let the function $f$ be defined by f(x)=4x+6. Then $f(4)=$ [ANS] and $f(5)=$ [ANS]", "answer_v3": [ "22", "26" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0154", "subject": "Algebra", "topic": "Functions", "subtopic": "Function notation", "level": "3", "keywords": [ "functions" ], "problem_v1": "Evaluate the function $ h(t)=18-4t$ for $t=3 u$. Simplify your answer as much as possible.\n$h(3 u)=$ [ANS]", "answer_v1": [ "18-12*u" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Evaluate the function $ h(t)=10-5t$ for $t=2 u$. Simplify your answer as much as possible.\n$h(2 u)=$ [ANS]", "answer_v2": [ "10-10*u" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Evaluate the function $ h(t)=13-4t$ for $t=3 u$. Simplify your answer as much as possible.\n$h(3 u)=$ [ANS]", "answer_v3": [ "13-12*u" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0155", "subject": "Algebra", "topic": "Functions", "subtopic": "Function notation", "level": "2", "keywords": [ "functions" ], "problem_v1": "Suppose a function $h$ is given by the following table of values.\n$\\begin{array}{cccccccc}\\hline t=&-3 &-2 &-1 & 0 & 1 & 2 & 3 \\\\ \\hline h(t)=&-2 &-2 &-1 &-4 &-1 & 0 & 0 \\\\ \\hline \\end{array}$\nUse the table to fill in the missing values. There may be more than one correct answer, in which case you should enter your answers as a comma separated list. If there are no correct answers, enter NONE. $h \\big($ [ANS] $\\big)=2h(-2)$ $h \\big($ [ANS] $\\big)=2h(1)+h(-3)$ $h \\big($ [ANS] $\\big)=h(2)$ $h \\big($ [ANS] $\\big)=h(-1)+h(-3)$", "answer_v1": [ "0", "0", "(2, 3)", "NONE" ], "answer_type_v1": [ "NV", "NV", "UOL", "OE" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Suppose a function $h$ is given by the following table of values.\n$\\begin{array}{cccccccc}\\hline t=&-3 &-2 &-1 & 0 & 1 & 2 & 3 \\\\ \\hline h(t)=&-1 & 0 & 0 &-2 &-2 &-4 &-1 \\\\ \\hline \\end{array}$\nUse the table to fill in the missing values. There may be more than one correct answer, in which case you should enter your answers as a comma separated list. If there are no correct answers, enter NONE. $h \\big($ [ANS] $\\big)=2h(0)$ $h \\big($ [ANS] $\\big)=2h(-3)+h(1)$ $h \\big($ [ANS] $\\big)=h(-1)$ $h \\big($ [ANS] $\\big)=h(3)+h(1)$", "answer_v2": [ "2", "2", "(-1, -2)", "NONE" ], "answer_type_v2": [ "NV", "NV", "UOL", "OE" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Suppose a function $h$ is given by the following table of values.\n$\\begin{array}{cccccccc}\\hline t=&-3 &-2 &-1 & 0 & 1 & 2 & 3 \\\\ \\hline h(t)=&-4 &-1 & 0 &-2 &-1 &-2 & 0 \\\\ \\hline \\end{array}$\nUse the table to fill in the missing values. There may be more than one correct answer, in which case you should enter your answers as a comma separated list. If there are no correct answers, enter NONE. $h \\big($ [ANS] $\\big)=2h(2)$ $h \\big($ [ANS] $\\big)=2h(1)+h(0)$ $h \\big($ [ANS] $\\big)=h(-1)$ $h \\big($ [ANS] $\\big)=h(-2)+h(0)$", "answer_v3": [ "-3", "-3", "(-1, 3)", "NONE" ], "answer_type_v3": [ "NV", "NV", "UOL", "OE" ], "options_v3": [ [], [], [], [] ] }, { "id": "Algebra_0156", "subject": "Algebra", "topic": "Functions", "subtopic": "Function notation", "level": "3", "keywords": [], "problem_v1": "Let $f(t)=5-3t^{2}.$ Evaluate $f(t+1)$.\n$f(t+1)=$ [ANS]\nNote: Your answer should be completely simplified. Unsimplified answers will not be accepted.", "answer_v1": [ "5-3*(t+1)^2" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Let $f(t)=2-3t^{2}.$ Evaluate $f(t+1)$.\n$f(t+1)=$ [ANS]\nNote: Your answer should be completely simplified. Unsimplified answers will not be accepted.", "answer_v2": [ "2-3*(t+1)^2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Let $f(t)=3-3t^{2}.$ Evaluate $f(t+1)$.\n$f(t+1)=$ [ANS]\nNote: Your answer should be completely simplified. Unsimplified answers will not be accepted.", "answer_v3": [ "3-3*(t+1)^2" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0157", "subject": "Algebra", "topic": "Functions", "subtopic": "Function notation", "level": "2", "keywords": [ "algebra", "function", "Functions", "Evaluation" ], "problem_v1": "List all real values of $x$ such that $f(x)=0$. If there are no such real $x$, type DNE in the answer blank. If there is more that one real $x$, give a comma separated list (i.e.: 1,2).\nf(x)=\\frac {10x+3}{5} $x$=[ANS]", "answer_v1": [ "-0.3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "List all real values of $x$ such that $f(x)=0$. If there are no such real $x$, type DNE in the answer blank. If there is more that one real $x$, give a comma separated list (i.e.: 1,2).\nf(x)=\\frac {-17x+18}{-14} $x$=[ANS]", "answer_v2": [ "1.05882352941176" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "List all real values of $x$ such that $f(x)=0$. If there are no such real $x$, type DNE in the answer blank. If there is more that one real $x$, give a comma separated list (i.e.: 1,2).\nf(x)=\\frac {-8x+4}{-9} $x$=[ANS]", "answer_v3": [ "0.5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0158", "subject": "Algebra", "topic": "Functions", "subtopic": "Graphs", "level": "2", "keywords": [ "calculus", "intervals", "absolute value", "functions" ], "problem_v1": "Find the interval on which the funtion $f(x)=|x+8|$ is increasing: [ANS]", "answer_v1": [ "(-8,infinity)" ], "answer_type_v1": [ "INT" ], "options_v1": [ [] ], "problem_v2": "Find the interval on which the funtion $f(x)=|x+1|$ is increasing: [ANS]", "answer_v2": [ "(-1,infinity)" ], "answer_type_v2": [ "INT" ], "options_v2": [ [] ], "problem_v3": "Find the interval on which the funtion $f(x)=|x+4|$ is increasing: [ANS]", "answer_v3": [ "(-4,infinity)" ], "answer_type_v3": [ "INT" ], "options_v3": [ [] ] }, { "id": "Algebra_0159", "subject": "Algebra", "topic": "Functions", "subtopic": "Graphs", "level": "2", "keywords": [ "Algebra", "Rational Functions" ], "problem_v1": "Consider the function f(x)=\\frac{x}{\\sqrt{16-x^2}} a) Determine the domain of the function. Note: Write the answer in interval notation. If the answer involves more than one interval write the intervals separated by the union symbol, U. If needed enter $\\infty$ as infinity and $-\\infty$ as-infinity. Domain=[ANS]\nb) Find the vertical asymptote(s). If there is more than one vertical asymptote give a list of the $x$-values separated by commas. If there are no vertical asymptotes type in None. $x=$ [ANS]", "answer_v1": [ "(-4,4)", "(-4, 4)" ], "answer_type_v1": [ "INT", "INT" ], "options_v1": [ [], [] ], "problem_v2": "Consider the function f(x)=\\frac{x}{\\sqrt{1-x^2}} a) Determine the domain of the function. Note: Write the answer in interval notation. If the answer involves more than one interval write the intervals separated by the union symbol, U. If needed enter $\\infty$ as infinity and $-\\infty$ as-infinity. Domain=[ANS]\nb) Find the vertical asymptote(s). If there is more than one vertical asymptote give a list of the $x$-values separated by commas. If there are no vertical asymptotes type in None. $x=$ [ANS]", "answer_v2": [ "(-1,1)", "(-1, 1)" ], "answer_type_v2": [ "INT", "INT" ], "options_v2": [ [], [] ], "problem_v3": "Consider the function f(x)=\\frac{x}{\\sqrt{4-x^2}} a) Determine the domain of the function. Note: Write the answer in interval notation. If the answer involves more than one interval write the intervals separated by the union symbol, U. If needed enter $\\infty$ as infinity and $-\\infty$ as-infinity. Domain=[ANS]\nb) Find the vertical asymptote(s). If there is more than one vertical asymptote give a list of the $x$-values separated by commas. If there are no vertical asymptotes type in None. $x=$ [ANS]", "answer_v3": [ "(-2,2)", "(-2, 2)" ], "answer_type_v3": [ "INT", "INT" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0160", "subject": "Algebra", "topic": "Functions", "subtopic": "Graphs", "level": "2", "keywords": [ "algebra", "functions", "graphing calculator" ], "problem_v1": "Use a graphing calculator to decide which viewing rectangle (A)-(D) produces the most appropriate graph of the equation. y=\\sqrt[4]{1296-x^2} Choose one: [ANS] A. [-10,10] by [-2,8] B. [-10,10] by [-10,10] C. [-36,36] by [-2,8] D. [0,36] by [-2,8]\nNote: The answers are given by [Xmin, Xmax] by [Ymin, Ymax]", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "Use a graphing calculator to decide which viewing rectangle (A)-(D) produces the most appropriate graph of the equation. y=\\sqrt[4]{256-x^2} Choose one: [ANS] A. [-10,10] by [-2,6] B. [-10,10] by [-10,10] C. [0,16] by [-2,6] D. [-16,16] by [-2,6]\nNote: The answers are given by [Xmin, Xmax] by [Ymin, Ymax]", "answer_v2": [ "D" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "Use a graphing calculator to decide which viewing rectangle (A)-(D) produces the most appropriate graph of the equation. y=\\sqrt[4]{256-x^2} Choose one: [ANS] A. [-10,10] by [-2,6] B. [-16,16] by [-2,6] C. [0,16] by [-2,6] D. [-10,10] by [-10,10]\nNote: The answers are given by [Xmin, Xmax] by [Ymin, Ymax]", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Algebra_0161", "subject": "Algebra", "topic": "Functions", "subtopic": "Domain and range", "level": "3", "keywords": [ "domain", "function' 'graph" ], "problem_v1": "Determine which of the following statements are true and which are false. Enter the T or F in front of each statement. Remember that $x\\in(-1,1)$ is the same as $-1 < x < 1$ and $x\\in[-1,1]$ means $-1 \\le x \\le 1$. [ANS] 1. The function $\\sin(x)$ on the domain $x\\in[-\\pi,\\pi]$ has at least one input which produces a largest output value. [ANS] 2. The function $\\sin(x)$ on the domain $x\\in(-\\pi,\\pi)$ has at least one input which produces a largest output value. [ANS] 3. The function $f(x)=x^3$ with domain $x\\in[-3,3]$ has at least one input which produces a smallest output value. [ANS] 4. The function $\\sin(x)$ on the domain $x\\in(-\\pi,\\pi)$ has at least one input which produces a smallest output value. [ANS] 5. The function $f(x)=x^3$ with domain $x\\in(-3,3)$ has at least one input which produces a smallest output value.", "answer_v1": [ "T", "T", "T", "T", "F" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Determine which of the following statements are true and which are false. Enter the T or F in front of each statement. Remember that $x\\in(-1,1)$ is the same as $-1 < x < 1$ and $x\\in[-1,1]$ means $-1 \\le x \\le 1$. [ANS] 1. The function $f(x)=x^3$ with domain $x\\in(-3,3)$ has at least one input which produces a largest output value. [ANS] 2. The function $\\sin(x)$ on the domain $x\\in[-\\pi,\\pi]$ has at least one input which produces a smallest output value. [ANS] 3. The function $f(x)=x^3$ with domain $x\\in(-3,3)$ has at least one input which produces a smallest output value. [ANS] 4. The function $f(x)=x^3$ with domain $x\\in[-3,3]$ has at least one input which produces a smallest output value. [ANS] 5. The function $\\sin(x)$ on the domain $x\\in[-\\pi,\\pi]$ has at least one input which produces a largest output value.", "answer_v2": [ "F", "T", "F", "T", "T" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Determine which of the following statements are true and which are false. Enter the T or F in front of each statement. Remember that $x\\in(-1,1)$ is the same as $-1 < x < 1$ and $x\\in[-1,1]$ means $-1 \\le x \\le 1$. [ANS] 1. The function $f(x)=x^3$ with domain $x\\in[-3,3]$ has at least one input which produces a largest output value. [ANS] 2. The function $\\sin(x)$ on the domain $x\\in(-\\pi,\\pi)$ has at least one input which produces a smallest output value. [ANS] 3. The function $f(x)=x^3$ with domain $x\\in(-3,3)$ has at least one input which produces a smallest output value. [ANS] 4. The function $\\sin(x)$ on the domain $x\\in(-\\pi,\\pi)$ has at least one input which produces a largest output value. [ANS] 5. The function $f(x)=x^3$ with domain $x\\in(-3,3)$ has at least one input which produces a largest output value.", "answer_v3": [ "T", "T", "F", "T", "F" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Algebra_0162", "subject": "Algebra", "topic": "Functions", "subtopic": "Domain and range", "level": "2", "keywords": [ "functions", "polynomials", "domain" ], "problem_v1": "The domain of the function $ f(x)=x+\\frac{7}{3x-15}$ is all real numbers $x$ except for where $x$ equals [ANS]", "answer_v1": [ "15/3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The domain of the function $ f(x)=x+\\frac{1}{4x-8}$ is all real numbers $x$ except for where $x$ equals [ANS]", "answer_v2": [ "8/4" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The domain of the function $ f(x)=x+\\frac{3}{3x-9}$ is all real numbers $x$ except for where $x$ equals [ANS]", "answer_v3": [ "9/3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0163", "subject": "Algebra", "topic": "Functions", "subtopic": "Piecewise functions", "level": "2", "keywords": [], "problem_v1": "$\\lceil 2.6\\rceil=$ [ANS]\n$\\lceil 0.800000000000001\\rceil=$ [ANS]\n$\\lceil 1.2\\rceil=$ [ANS]\n$\\lceil 2.2\\rceil=$ [ANS]\n$\\lceil-2\\rceil=$ [ANS]", "answer_v1": [ "3", "1", "2", "3", "-2" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "$\\lceil-4.2\\rceil=$ [ANS]\n$\\lceil 4.4\\rceil=$ [ANS]\n$\\lceil-3.5\\rceil=$ [ANS]\n$\\lceil-1.7\\rceil=$ [ANS]\n$\\lceil 4.5\\rceil=$ [ANS]", "answer_v2": [ "-4", "5", "-3", "-1", "5" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "$\\lceil-1.9\\rceil=$ [ANS]\n$\\lceil 1.1\\rceil=$ [ANS]\n$\\lceil-2.2\\rceil=$ [ANS]\n$\\lceil 0.5\\rceil=$ [ANS]\n$\\lceil-3\\rceil=$ [ANS]", "answer_v3": [ "-1", "2", "-2", "1", "-3" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Algebra_0164", "subject": "Algebra", "topic": "Functions", "subtopic": "Piecewise functions", "level": "2", "keywords": [ "piecewise functions", "piecewise", "functions", "applications" ], "problem_v1": "A mobile plan charges a base monthly fee of \\$25.00 for the first 500 minutes of air time plus a charge of \\$0.65 for each additional minute. Write a piecewise-defined linear function which calculates the monthly cost $C$ (in dollars) for using $m$ minutes of air time.\n$\\begin{array}{cccc}\\hline & C(m)=\\left\\lbrace \\begin{array}{cc} &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ \\end{array}\\right. & & [ANS]if [ANS] \\leq m \\leq [ANS] [ANS]if m > [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "25", "0", "500", "25+0.65*(m-500)", "500" ], "answer_type_v1": [ "NV", "NV", "NV", "EX", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "A mobile plan charges a base monthly fee of \\$10.00 for the first 650 minutes of air time plus a charge of \\$0.20 for each additional minute. Write a piecewise-defined linear function which calculates the monthly cost $C$ (in dollars) for using $m$ minutes of air time.\n$\\begin{array}{cccc}\\hline & C(m)=\\left\\lbrace \\begin{array}{cc} &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ \\end{array}\\right. & & [ANS]if [ANS] \\leq m \\leq [ANS] [ANS]if m > [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "10", "0", "650", "10+0.2*(m-650)", "650" ], "answer_type_v2": [ "NV", "NV", "NV", "EX", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "A mobile plan charges a base monthly fee of \\$15.00 for the first 500 minutes of air time plus a charge of \\$0.35 for each additional minute. Write a piecewise-defined linear function which calculates the monthly cost $C$ (in dollars) for using $m$ minutes of air time.\n$\\begin{array}{cccc}\\hline & C(m)=\\left\\lbrace \\begin{array}{cc} &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ \\end{array}\\right. & & [ANS]if [ANS] \\leq m \\leq [ANS] [ANS]if m > [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "15", "0", "500", "15+0.35*(m-500)", "500" ], "answer_type_v3": [ "NV", "NV", "NV", "EX", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Algebra_0165", "subject": "Algebra", "topic": "Functions", "subtopic": "Piecewise functions", "level": "2", "keywords": [ "piecewise functions", "piecewise", "functions", "applications" ], "problem_v1": "A restaurant offers a catering service which costs \\$22.50 per person with a \\$117.00 service charge. For parties of 60 or more people, a group discount applies, and the cost is \\$18.50 per person along with the service charge dropping to \\$58.00. Write a piecewise-defined linear function which calculates the total cost $T$ of the catering service which serves $n$ people.\n$\\begin{array}{cccc}\\hline & T(n)=\\left\\lbrace \\begin{array}{cc} &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ \\end{array}\\right. & & [ANS]if [ANS] \\leq n \\leq [ANS] [ANS]if n \\geq [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "22.5*n+117", "1", "60-1", "18.5*n+58", "60" ], "answer_type_v1": [ "EX", "NV", "NV", "EX", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "A restaurant offers a catering service which costs \\$15.50 per person with a \\$74.25 service charge. For parties of 75 or more people, a group discount applies, and the cost is \\$12.75 per person along with the service charge dropping to \\$37.00. Write a piecewise-defined linear function which calculates the total cost $T$ of the catering service which serves $n$ people.\n$\\begin{array}{cccc}\\hline & T(n)=\\left\\lbrace \\begin{array}{cc} &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ \\end{array}\\right. & & [ANS]if [ANS] \\leq n \\leq [ANS] [ANS]if n \\geq [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "15.5*n+74.25", "1", "75-1", "12.75*n+37", "75" ], "answer_type_v2": [ "EX", "NV", "NV", "EX", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "A restaurant offers a catering service which costs \\$18.00 per person with a \\$91.00 service charge. For parties of 60 or more people, a group discount applies, and the cost is \\$14.75 per person along with the service charge dropping to \\$45.00. Write a piecewise-defined linear function which calculates the total cost $T$ of the catering service which serves $n$ people.\n$\\begin{array}{cccc}\\hline & T(n)=\\left\\lbrace \\begin{array}{cc} &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ \\end{array}\\right. & & [ANS]if [ANS] \\leq n \\leq [ANS] [ANS]if n \\geq [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "18*n+91", "1", "60-1", "14.75*n+45", "60" ], "answer_type_v3": [ "EX", "NV", "NV", "EX", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Algebra_0166", "subject": "Algebra", "topic": "Functions", "subtopic": "Piecewise functions", "level": "", "keywords": [ "piecewise functions", "piecewise", "functions", "applications" ], "problem_v1": "A local pet shop charges \\$0.76 per cricket up to 150 crickets, and \\$0.69 per cricket thereafter. Write a piecewise-defined linear function which calculates the price $P$, in dollars, of purchasing $c$ crickets.\n$\\begin{array}{cccc}\\hline & P(c)=\\left\\lbrace \\begin{array}{cc} &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ \\end{array}\\right. & & [ANS]if [ANS] \\leq c \\leq [ANS] [ANS]if c > [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "0.76*c", "1", "150", "114+0.69*(c-150)", "150" ], "answer_type_v1": [ "EX", "NV", "NV", "EX", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "A local pet shop charges \\$0.16 per cricket up to 195 crickets, and \\$0.13 per cricket thereafter. Write a piecewise-defined linear function which calculates the price $P$, in dollars, of purchasing $c$ crickets.\n$\\begin{array}{cccc}\\hline & P(c)=\\left\\lbrace \\begin{array}{cc} &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ \\end{array}\\right. & & [ANS]if [ANS] \\leq c \\leq [ANS] [ANS]if c > [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "0.16*c", "1", "195", "31.2+0.13*(c-195)", "195" ], "answer_type_v2": [ "EX", "NV", "NV", "EX", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "A local pet shop charges \\$0.38 per cricket up to 155 crickets, and \\$0.34 per cricket thereafter. Write a piecewise-defined linear function which calculates the price $P$, in dollars, of purchasing $c$ crickets.\n$\\begin{array}{cccc}\\hline & P(c)=\\left\\lbrace \\begin{array}{cc} &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ &\\\\ \\end{array}\\right. & & [ANS]if [ANS] \\leq c \\leq [ANS] [ANS]if c > [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "0.38*c", "1", "155", "58.9+0.34*(c-155)", "155" ], "answer_type_v3": [ "EX", "NV", "NV", "EX", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Algebra_0167", "subject": "Algebra", "topic": "Functions", "subtopic": "Compositions and combinations of functions", "level": "1", "keywords": [ "composition Of Functions" ], "problem_v1": "Consider the functions\n$f(x)=7 ^x,$ $g(x)=x ^{7}$\nDetermine the following compositions of functions:\n$(f \\circ f)(x)=$ [ANS]\n$(f \\circ g)(x)=$ [ANS]\n$(g \\circ f)(x)=$ [ANS]\n$(g \\circ g)(x)=$ [ANS]", "answer_v1": [ "7^(7^x)", "7^(x^7)", "7^(7*x)", "x^(7*7)" ], "answer_type_v1": [ "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Consider the functions\n$f(x)=3 ^x,$ $g(x)=x ^{3}$\nDetermine the following compositions of functions:\n$(f \\circ f)(x)=$ [ANS]\n$(f \\circ g)(x)=$ [ANS]\n$(g \\circ f)(x)=$ [ANS]\n$(g \\circ g)(x)=$ [ANS]", "answer_v2": [ "3^(3^x)", "3^(x^3)", "3^(3*x)", "x^(3*3)" ], "answer_type_v2": [ "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Consider the functions\n$f(x)=4 ^x,$ $g(x)=x ^{4}$\nDetermine the following compositions of functions:\n$(f \\circ f)(x)=$ [ANS]\n$(f \\circ g)(x)=$ [ANS]\n$(g \\circ f)(x)=$ [ANS]\n$(g \\circ g)(x)=$ [ANS]", "answer_v3": [ "4^(4^x)", "4^(x^4)", "4^(4*x)", "x^(4*4)" ], "answer_type_v3": [ "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Algebra_0168", "subject": "Algebra", "topic": "Functions", "subtopic": "Compositions and combinations of functions", "level": "3", "keywords": [ "composition", "function" ], "problem_v1": "Let $u(x)=e^{8x}$ and $v(x)=6x+6$. Find a simplified formula for the function below. $v \\big(u(x)^2 \\big)=$ [ANS] $+$ [ANS]", "answer_v1": [ "6*e^(16*x)", "6" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $u(x)=e^{2x}$ and $v(x)=9x+3$. Find a simplified formula for the function below. $v \\big(u(x)^2 \\big)=$ [ANS] $+$ [ANS]", "answer_v2": [ "9*e^(4*x)", "3" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $u(x)=e^{4x}$ and $v(x)=6x+4$. Find a simplified formula for the function below. $v \\big(u(x)^2 \\big)=$ [ANS] $+$ [ANS]", "answer_v3": [ "6*e^(8*x)", "4" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0169", "subject": "Algebra", "topic": "Functions", "subtopic": "Compositions and combinations of functions", "level": "2", "keywords": [ "composition", "function" ], "problem_v1": "Let $m(x)=4x^2-7$, $n(x)=5x$, and $o(x)=\\sqrt{3x+5}$. Find a simplified formula for the function below. Your final answer should be a polynomial with only one term in each power of $x$. $m(o(x)) n(x)=$ [ANS]", "answer_v1": [ "60*x^2+65*x" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Let $m(x)=2x^2-8$, $n(x)=2x$, and $o(x)=\\sqrt{2x+8}$. Find a simplified formula for the function below. Your final answer should be a polynomial with only one term in each power of $x$. $m(o(x)) n(x)=$ [ANS]", "answer_v2": [ "8*x^2+16*x" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Let $m(x)=2x^2-7$, $n(x)=3x$, and $o(x)=\\sqrt{3x+5}$. Find a simplified formula for the function below. Your final answer should be a polynomial with only one term in each power of $x$. $m(o(x)) n(x)=$ [ANS]", "answer_v3": [ "18*x^2+9*x" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0170", "subject": "Algebra", "topic": "Functions", "subtopic": "Compositions and combinations of functions", "level": "2", "keywords": [ "composition", "decomposition", "function" ], "problem_v1": "Use the table of values for the functions $p(x)$ and $q(x)$ below to complete the tables for the composite functions defined in parts\n(a) and (b):\n$\\begin{array}{ccccccc}\\hline x & 0 & 1 & 2 & 3 & 4 & 5 \\\\ \\hline p(x) & 4 & 2 & 3 & 5 & 0 & 1 \\\\ \\hline q(x) & 3 & 2 & 1 & 4 & 5 & 0 \\\\ \\hline \\end{array}$\n(a) Complete the table of values for the composite function $r(x)=p(q(x))$ at $x=0, 1, 2, 3, 4, 5$\n$\\begin{array}{ccccccc}\\hline x & 0 & 1 & 2 & 3 & 4 & 5 \\\\ \\hline r(x) & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\n(b) Complete the table of values for the composite function $s(x)=q(p(x))$ at $x=0, 1, 2, 3, 4, 5$.\n$\\begin{array}{ccccccc}\\hline x & 0 & 1 & 2 & 3 & 4 & 5 \\\\ \\hline s(x) & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "5", "3", "2", "0", "1", "4", "5", "1", "4", "0", "3", "2" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Use the table of values for the functions $p(x)$ and $q(x)$ below to complete the tables for the composite functions defined in parts\n(a) and (b):\n$\\begin{array}{ccccccc}\\hline x & 0 & 1 & 2 & 3 & 4 & 5 \\\\ \\hline p(x) & 0 & 5 & 1 & 3 & 4 & 2 \\\\ \\hline q(x) & 1 & 2 & 4 & 0 & 5 & 3 \\\\ \\hline \\end{array}$\n(a) Complete the table of values for the composite function $r(x)=p(q(x))$ at $x=0, 1, 2, 3, 4, 5$\n$\\begin{array}{ccccccc}\\hline x & 0 & 1 & 2 & 3 & 4 & 5 \\\\ \\hline r(x) & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\n(b) Complete the table of values for the composite function $s(x)=q(p(x))$ at $x=0, 1, 2, 3, 4, 5$.\n$\\begin{array}{ccccccc}\\hline x & 0 & 1 & 2 & 3 & 4 & 5 \\\\ \\hline s(x) & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "5", "1", "4", "0", "2", "3", "1", "3", "2", "0", "5", "4" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Use the table of values for the functions $p(x)$ and $q(x)$ below to complete the tables for the composite functions defined in parts\n(a) and (b):\n$\\begin{array}{ccccccc}\\hline x & 0 & 1 & 2 & 3 & 4 & 5 \\\\ \\hline p(x) & 1 & 4 & 2 & 3 & 0 & 5 \\\\ \\hline q(x) & 4 & 5 & 3 & 0 & 1 & 2 \\\\ \\hline \\end{array}$\n(a) Complete the table of values for the composite function $r(x)=p(q(x))$ at $x=0, 1, 2, 3, 4, 5$\n$\\begin{array}{ccccccc}\\hline x & 0 & 1 & 2 & 3 & 4 & 5 \\\\ \\hline r(x) & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\n(b) Complete the table of values for the composite function $s(x)=q(p(x))$ at $x=0, 1, 2, 3, 4, 5$.\n$\\begin{array}{ccccccc}\\hline x & 0 & 1 & 2 & 3 & 4 & 5 \\\\ \\hline s(x) & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "0", "5", "3", "1", "4", "2", "5", "1", "3", "0", "4", "2" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Algebra_0171", "subject": "Algebra", "topic": "Functions", "subtopic": "Compositions and combinations of functions", "level": "3", "keywords": [ "functions", "domain", "range", "input", "output", "interval notation" ], "problem_v1": "The formula for the volume of a cube with side length $s$ is $V=s^3$. The formula for the surface area of a cube is $A=6s^2$.\n(a) Find the formula for the function $s=f(A)$. $s=f(A)=$ [ANS]\nWhich of the statements best explains the meaning of $s=f(A)$? [ANS] A. The side length for a cube of surface area $A$ B. The volume of a cube of side length $s$ C. The side length for a cube of volume $V$ D. The surface area of a cube of side length $s$\n(b) If $V=g(s)$, find a formula for $g(f(A)).$ $g(f(A))=$ [ANS]\nWhich of the statements best explains the meaning of $g(f(A))$? [ANS] A. The volume for a cube with surface area $A$ B. The surface area for a cube of side length $s$ C. The surface area for a cube of volume $V$ D. The volume for a cube of side length $s$", "answer_v1": [ "sqrt(A/6)", "A", "(A/6)^1.5", "A" ], "answer_type_v1": [ "EX", "MCS", "EX", "MCS" ], "options_v1": [ [], [ "A", "B", "C", "D" ], [], [ "A", "B", "C", "D" ] ], "problem_v2": "The formula for the volume of a cube with side length $s$ is $V=s^3$. The formula for the surface area of a cube is $A=6s^2$.\n(a) Find the formula for the function $s=f(A)$. $s=f(A)=$ [ANS]\nWhich of the statements best explains the meaning of $s=f(A)$? [ANS] A. The side length for a cube of surface area $A$ B. The volume of a cube of side length $s$ C. The side length for a cube of volume $V$ D. The surface area of a cube of side length $s$\n(b) If $V=g(s)$, find a formula for $g(f(A)).$ $g(f(A))=$ [ANS]\nWhich of the statements best explains the meaning of $g(f(A))$? [ANS] A. The surface area for a cube of volume $V$ B. The surface area for a cube of side length $s$ C. The volume for a cube of side length $s$ D. The volume for a cube with surface area $A$", "answer_v2": [ "sqrt(A/6)", "A", "(A/6)^1.5", "D" ], "answer_type_v2": [ "EX", "MCS", "EX", "MCS" ], "options_v2": [ [], [ "A", "B", "C", "D" ], [], [ "A", "B", "C", "D" ] ], "problem_v3": "The formula for the volume of a cube with side length $s$ is $V=s^3$. The formula for the surface area of a cube is $A=6s^2$.\n(a) Find the formula for the function $s=f(A)$. $s=f(A)=$ [ANS]\nWhich of the statements best explains the meaning of $s=f(A)$? [ANS] A. The side length for a cube of surface area $A$ B. The side length for a cube of volume $V$ C. The volume of a cube of side length $s$ D. The surface area of a cube of side length $s$\n(b) If $V=g(s)$, find a formula for $g(f(A)).$ $g(f(A))=$ [ANS]\nWhich of the statements best explains the meaning of $g(f(A))$? [ANS] A. The surface area for a cube of volume $V$ B. The volume for a cube with surface area $A$ C. The surface area for a cube of side length $s$ D. The volume for a cube of side length $s$", "answer_v3": [ "sqrt(A/6)", "A", "(A/6)^1.5", "B" ], "answer_type_v3": [ "EX", "MCS", "EX", "MCS" ], "options_v3": [ [], [ "A", "B", "C", "D" ], [], [ "A", "B", "C", "D" ] ] }, { "id": "Algebra_0172", "subject": "Algebra", "topic": "Functions", "subtopic": "Compositions and combinations of functions", "level": "3", "keywords": [ "domain", "function' 'composition", "functions", "polynomials", "composition" ], "problem_v1": "Let $f(x)=\\sqrt{56-x}$ and $g(x)=x^2-x$. Then the domain of $f\\circ g$ is equal to $[a,b]$ for $a=$ [ANS]\nand $b=$ [ANS]", "answer_v1": [ "-7", "8" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $f(x)=\\sqrt{2-x}$ and $g(x)=x^2-x$. Then the domain of $f\\circ g$ is equal to $[a,b]$ for $a=$ [ANS]\nand $b=$ [ANS]", "answer_v2": [ "-1", "2" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $f(x)=\\sqrt{12-x}$ and $g(x)=x^2-x$. Then the domain of $f\\circ g$ is equal to $[a,b]$ for $a=$ [ANS]\nand $b=$ [ANS]", "answer_v3": [ "-3", "4" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0173", "subject": "Algebra", "topic": "Functions", "subtopic": "Compositions and combinations of functions", "level": "2", "keywords": [ "compose", "composition", "decomposition" ], "problem_v1": "Use substitution to compose $y=8 u^2+4 u+4$ and $u=4x^{2}$. Enter your answer as an equation, and simplify your answer as much as possible. [ANS]", "answer_v1": [ "y = 128*x^4+16*x^2+4" ], "answer_type_v1": [ "EQ" ], "options_v1": [ [] ], "problem_v2": "Use substitution to compose $y=2 u^2+5 u+2$ and $u=3x^{4}$. Enter your answer as an equation, and simplify your answer as much as possible. [ANS]", "answer_v2": [ "y = 18*x^8+15*x^4+2" ], "answer_type_v2": [ "EQ" ], "options_v2": [ [] ], "problem_v3": "Use substitution to compose $y=4 u^2+4 u+3$ and $u=4x^{2}$. Enter your answer as an equation, and simplify your answer as much as possible. [ANS]", "answer_v3": [ "y = 64*x^4+16*x^2+3" ], "answer_type_v3": [ "EQ" ], "options_v3": [ [] ] }, { "id": "Algebra_0174", "subject": "Algebra", "topic": "Functions", "subtopic": "Compositions and combinations of functions", "level": "3", "keywords": [ "compose", "composition", "decomposition" ], "problem_v1": "Write the function $y=\\sqrt{8-x^{6}}$ in the form $y=k \\cdot (h(x))^p$ for some function $h(x)$ and some constants $k$ and $p$.\n$k$=[ANS]\n$p$=[ANS]\n$h(x)$=[ANS]", "answer_v1": [ "1", "0.5", "8-x^6" ], "answer_type_v1": [ "NV", "NV", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Write the function $y=\\sqrt{2-x^{9}}$ in the form $y=k \\cdot (h(x))^p$ for some function $h(x)$ and some constants $k$ and $p$.\n$k$=[ANS]\n$p$=[ANS]\n$h(x)$=[ANS]", "answer_v2": [ "1", "0.5", "2-x^9" ], "answer_type_v2": [ "NV", "NV", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Write the function $y=\\sqrt{4-x^{6}}$ in the form $y=k \\cdot (h(x))^p$ for some function $h(x)$ and some constants $k$ and $p$.\n$k$=[ANS]\n$p$=[ANS]\n$h(x)$=[ANS]", "answer_v3": [ "1", "0.5", "4-x^6" ], "answer_type_v3": [ "NV", "NV", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0175", "subject": "Algebra", "topic": "Functions", "subtopic": "Difference quotient", "level": "3", "keywords": [ "functions", "concavity", "increasing", "decreasing" ], "problem_v1": "Calculate the successive average rates of change for the function, $H(x)$ in the table below.\n$\\begin{array}{ccccc}\\hline x & 12 & 16 & 20 & 24 \\\\ \\hline H(x) & 21.5 & 21.64 & 21.85 & 22.16 \\\\ \\hline \\end{array}$\n(a) The average rate of change over the interval $12 \\leq x \\leq 16$ is [ANS]\n(Retain at least 3 decimal places in your answer.) (Retain at least 3 decimal places in your answer.) (b) The average rate of change over the interval $16 \\leq x \\leq 20$ is [ANS]\n(Retain at least 3 decimal places in your answer.) (Retain at least 3 decimal places in your answer.) (c) The average rate of change over the interval $20 \\leq x \\leq 24$ is [ANS]\n(Retain at least 3 decimal places in your answer.) (Retain at least 3 decimal places in your answer.) (d) Based your answers for the rates of change, the function $H(x)$ is [ANS] A. Concave Up B. Concave Down C. Neither concave up or concave down D. Both concave up and concave down", "answer_v1": [ "0.035", "0.0525", "0.0775", "A" ], "answer_type_v1": [ "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v2": "Calculate the successive average rates of change for the function, $H(x)$ in the table below.\n$\\begin{array}{ccccc}\\hline x & 12 & 14 & 16 & 18 \\\\ \\hline H(x) & 21.8 & 22.1 & 22.28 & 22.4 \\\\ \\hline \\end{array}$\n(a) The average rate of change over the interval $12 \\leq x \\leq 14$ is [ANS]\n(Retain at least 3 decimal places in your answer.) (Retain at least 3 decimal places in your answer.) (b) The average rate of change over the interval $14 \\leq x \\leq 16$ is [ANS]\n(Retain at least 3 decimal places in your answer.) (Retain at least 3 decimal places in your answer.) (c) The average rate of change over the interval $16 \\leq x \\leq 18$ is [ANS]\n(Retain at least 3 decimal places in your answer.) (Retain at least 3 decimal places in your answer.) (d) Based your answers for the rates of change, the function $H(x)$ is [ANS] A. Concave Up B. Concave Down C. Neither concave up or concave down D. Both concave up and concave down", "answer_v2": [ "0.15", "0.09", "0.06", "B" ], "answer_type_v2": [ "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v3": "Calculate the successive average rates of change for the function, $H(x)$ in the table below.\n$\\begin{array}{ccccc}\\hline x & 12 & 14 & 16 & 18 \\\\ \\hline H(x) & 21.5 & 21.81 & 21.99 & 22.13 \\\\ \\hline \\end{array}$\n(a) The average rate of change over the interval $12 \\leq x \\leq 14$ is [ANS]\n(Retain at least 3 decimal places in your answer.) (Retain at least 3 decimal places in your answer.) (b) The average rate of change over the interval $14 \\leq x \\leq 16$ is [ANS]\n(Retain at least 3 decimal places in your answer.) (Retain at least 3 decimal places in your answer.) (c) The average rate of change over the interval $16 \\leq x \\leq 18$ is [ANS]\n(Retain at least 3 decimal places in your answer.) (Retain at least 3 decimal places in your answer.) (d) Based your answers for the rates of change, the function $H(x)$ is [ANS] A. Concave Up B. Concave Down C. Neither concave up or concave down D. Both concave up and concave down", "answer_v3": [ "0.155", "0.09", "0.07", "B" ], "answer_type_v3": [ "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B", "C", "D" ] ] }, { "id": "Algebra_0176", "subject": "Algebra", "topic": "Functions", "subtopic": "Difference quotient", "level": "4", "keywords": [ "calculator", "slope", "tangent line" ], "problem_v1": "Let p(x)=4.9x^{1.9}. Use a calculator or a graphing program to find the slope of the tangent line to the point (x,p(x)) when x=3.1). Give the answer to 3 places. [ANS]", "answer_v1": [ "25.7736072011703" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let p(x)=1.8x^{0.9}. Use a calculator or a graphing program to find the slope of the tangent line to the point (x,p(x)) when x=4.7). Give the answer to 3 places. [ANS]", "answer_v2": [ "1.38773080447578" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let p(x)=2.9x^{1.1}. Use a calculator or a graphing program to find the slope of the tangent line to the point (x,p(x)) when x=3.2). Give the answer to 3 places. [ANS]", "answer_v3": [ "3.58348574244824" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0177", "subject": "Algebra", "topic": "Functions", "subtopic": "Difference quotient", "level": "3", "keywords": [ "average rate of change" ], "problem_v1": "(a) What is the average rate of change of $g(x)=2+5x$ between the points $(-3,-13)$ and $(5,27)$? answer=[ANS]\n(b) The function $g$ is [ANS] on the interval $-3 \\leq x \\leq 5$.", "answer_v1": [ "5", "increasing" ], "answer_type_v1": [ "NV", "MCS" ], "options_v1": [ [], [ "increasing", "decreasing" ] ], "problem_v2": "(a) What is the average rate of change of $g(x)=8-7x$ between the points $(-4, 36)$ and $(4,-20)$? answer=[ANS]\n(b) The function $g$ is [ANS] on the interval $-4 \\leq x \\leq 4$.", "answer_v2": [ "-7", "decreasing" ], "answer_type_v2": [ "NV", "MCS" ], "options_v2": [ [], [ "increasing", "decreasing" ] ], "problem_v3": "(a) What is the average rate of change of $g(x)=2-5x$ between the points $(-4, 22)$ and $(5,-23)$? answer=[ANS]\n(b) The function $g$ is [ANS] on the interval $-4 \\leq x \\leq 5$.", "answer_v3": [ "-5", "decreasing" ], "answer_type_v3": [ "NV", "MCS" ], "options_v3": [ [], [ "increasing", "decreasing" ] ] }, { "id": "Algebra_0178", "subject": "Algebra", "topic": "Functions", "subtopic": "Interpretation and applications", "level": "3", "keywords": [ "function", "domain", "range" ], "problem_v1": "Assume a car uses gas at a constant rate. After driving $20$ miles since a full tank of gas was purchased, there was $13$ gallons of gas left; after driving $55$ miles since a full tank of gas was purchased, there was $11.25$ gallons of gas left. Use a function to model the amount of gas in the tank (in gallons). Let the independent variable be the number of miles driven since a full tank of gas was purchased. Find this function\u2019s domain and range in this context.\nThe function\u2019s domain in this context is [ANS]. The function\u2019s range in this context is [ANS].", "answer_v1": [ "[0,280]", "[0,14]" ], "answer_type_v1": [ "INT", "INT" ], "options_v1": [ [], [] ], "problem_v2": "Assume a car uses gas at a constant rate. After driving $10$ miles since a full tank of gas was purchased, there was $8.7$ gallons of gas left; after driving $55$ miles since a full tank of gas was purchased, there was $7.35$ gallons of gas left. Use a function to model the amount of gas in the tank (in gallons). Let the independent variable be the number of miles driven since a full tank of gas was purchased. Find this function\u2019s domain and range in this context.\nThe function\u2019s domain in this context is [ANS]. The function\u2019s range in this context is [ANS].", "answer_v2": [ "[0,300]", "[0,9]" ], "answer_type_v2": [ "INT", "INT" ], "options_v2": [ [], [] ], "problem_v3": "Assume a car uses gas at a constant rate. After driving $30$ miles since a full tank of gas was purchased, there was $16.2$ gallons of gas left; after driving $55$ miles since a full tank of gas was purchased, there was $14.7$ gallons of gas left. Use a function to model the amount of gas in the tank (in gallons). Let the independent variable be the number of miles driven since a full tank of gas was purchased. Find this function\u2019s domain and range in this context.\nThe function\u2019s domain in this context is [ANS]. The function\u2019s range in this context is [ANS].", "answer_v3": [ "[0,300]", "[0,18]" ], "answer_type_v3": [ "INT", "INT" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0179", "subject": "Algebra", "topic": "Functions", "subtopic": "Interpretation and applications", "level": "5", "keywords": [ "function" ], "problem_v1": "The function $C$ models the number of customers that are in a store $t$ hours after the store opened on a certain day.\n$\\begin{array}{cccccccccccccc}\\hline t & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\\\ \\hline C(t) & 0 & 26 & 52 & 70 & 84 & 94 & 101 & 97 & 85 & 70 & 50 & 27 & 0 \\\\ \\hline \\end{array}$\n$C(9)=$ [ANS]\nInterpret the meaning of $C(9)$: [ANS] A. There were $9$ customers in the store $70$ hours after the store opened. [ANS] B. In $9$ hours since the store opened, there were a total of $70$ customers. C. There were $70$ customers in the store $9$ hours after the store opened. D. In $9$ hours since the store opened, the store had an average of $70$ customers per hour.\nSolve $C(t)=70$ for $t$. Use commas to separate your answers if there are more than one solution. $t=$ [ANS]\nInterpret the meaning of Part c\u2019s solution(s): [ANS] A. There were $70$ customers in the store $9$ hours after the store opened. [ANS] B. There were $70$ customers in the store either $3$ hours after the store opened, or $9$ hours after the store opened. C. There were $70$ customers in the store $3$ hours after the store opened, and again $9$ hours after the store opened. D. There were $70$ customers in the store $3$ hours after the store opened.", "answer_v1": [ "70", "C", "(3, 9)", "C" ], "answer_type_v1": [ "NV", "MCS", "UOL", "MCS" ], "options_v1": [ [], [ "A", "B", "C", "D" ], [], [ "A", "B", "C", "D" ] ], "problem_v2": "The function $C$ models the number of customers that are in a store $t$ hours after the store opened on a certain day.\n$\\begin{array}{cccccccccccccc}\\hline t & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\\\ \\hline C(t) & 0 & 21 & 48 & 75 & 84 & 96 & 97 & 96 & 84 & 71 & 45 & 27 & 0 \\\\ \\hline \\end{array}$\n$C(1)=$ [ANS]\nInterpret the meaning of $C(1)$: [ANS] A. There were $1$ customers in the store $21$ hours after the store opened. [ANS] B. In $1$ hours since the store opened, the store had an average of $21$ customers per hour. C. There were $21$ customers in the store $1$ hours after the store opened. D. In $1$ hours since the store opened, there were a total of $21$ customers.\nSolve $C(t)=96$ for $t$. Use commas to separate your answers if there are more than one solution. $t=$ [ANS]\nInterpret the meaning of Part c\u2019s solution(s): [ANS] A. There were $96$ customers in the store $5$ hours after the store opened, and again $7$ hours after the store opened. [ANS] B. There were $96$ customers in the store $7$ hours after the store opened. C. There were $96$ customers in the store $5$ hours after the store opened. D. There were $96$ customers in the store either $5$ hours after the store opened, or $7$ hours after the store opened.", "answer_v2": [ "21", "C", "(5, 7)", "A" ], "answer_type_v2": [ "NV", "MCS", "UOL", "MCS" ], "options_v2": [ [], [ "A", "B", "C", "D" ], [], [ "A", "B", "C", "D" ] ], "problem_v3": "The function $C$ models the number of customers that are in a store $t$ hours after the store opened on a certain day.\n$\\begin{array}{cccccccccccccc}\\hline t & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\\\ \\hline C(t) & 0 & 22 & 48 & 70 & 89 & 101 & 104 & 93 & 84 & 70 & 47 & 20 & 0 \\\\ \\hline \\end{array}$\n$C(4)=$ [ANS]\nInterpret the meaning of $C(4)$: [ANS] A. In $4$ hours since the store opened, there were a total of $89$ customers. [ANS] B. In $4$ hours since the store opened, the store had an average of $89$ customers per hour. C. There were $89$ customers in the store $4$ hours after the store opened. D. There were $4$ customers in the store $89$ hours after the store opened.\nSolve $C(t)=70$ for $t$. Use commas to separate your answers if there are more than one solution. $t=$ [ANS]\nInterpret the meaning of Part c\u2019s solution(s): [ANS] A. There were $70$ customers in the store $3$ hours after the store opened. [ANS] B. There were $70$ customers in the store $9$ hours after the store opened. C. There were $70$ customers in the store either $3$ hours after the store opened, or $9$ hours after the store opened. D. There were $70$ customers in the store $3$ hours after the store opened, and again $9$ hours after the store opened.", "answer_v3": [ "89", "C", "(3, 9)", "D" ], "answer_type_v3": [ "NV", "MCS", "UOL", "MCS" ], "options_v3": [ [], [ "A", "B", "C", "D" ], [], [ "A", "B", "C", "D" ] ] }, { "id": "Algebra_0180", "subject": "Algebra", "topic": "Functions", "subtopic": "Interpretation and applications", "level": "4", "keywords": [ "function" ], "problem_v1": "Rita will spend ${\\$150}$ to purchase some bowls and some plates. Each bowl costs ${\\$1}$, and each plate costs ${\\$6}$. The function $p(b)={-{\\textstyle\\frac{1}{6}}b+25}$ models the number of plates Rita to be purchase, where $b$ represents the number of bowls to be purchased. Interpret the meaning of $p(60)={15}$. [ANS] A. A. `15` bowls and `60` plates can be purchased. [ANS] B. B. `\\$15` will be used to purchase bowls, and `\\$60` will be used to purchase plates. C. C. `60` bowls and `15` plates can be purchased. D. D. `\\$60` will be used to purchase bowls, and `\\$15` will be used to purchase plates.", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "Hannah will spend ${\\$75}$ to purchase some bowls and some plates. Each bowl costs ${\\$2}$, and each plate costs ${\\$3}$. The function $p(b)={-{\\textstyle\\frac{2}{3}}b+25}$ models the number of plates Hannah to be purchase, where $b$ represents the number of bowls to be purchased. Interpret the meaning of $p(15)={15}$. [ANS] A. A. `15` bowls and `15` plates can be purchased. [ANS] B. B. `15` bowls and `15` plates can be purchased. C. C. `\\$15` will be used to purchase bowls, and `\\$15` will be used to purchase plates. D. D. `\\$15` will be used to purchase bowls, and `\\$15` will be used to purchase plates.", "answer_v2": [ "A" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "Ravi will spend ${\\$140}$ to purchase some bowls and some plates. Each bowl costs ${\\$1}$, and each plate costs ${\\$4}$. The function $p(b)={-{\\textstyle\\frac{1}{4}}b+35}$ models the number of plates Ravi to be purchase, where $b$ represents the number of bowls to be purchased. Interpret the meaning of $p(32)={27}$. [ANS] A. A. `\\$27` will be used to purchase bowls, and `\\$32` will be used to purchase plates. [ANS] B. B. `32` bowls and `27` plates can be purchased. C. C. `27` bowls and `32` plates can be purchased. D. D. `\\$32` will be used to purchase bowls, and `\\$27` will be used to purchase plates.", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Algebra_0181", "subject": "Algebra", "topic": "Functions", "subtopic": "Interpretation and applications", "level": "3", "keywords": [ "function" ], "problem_v1": "Suppose that $M$ is the function that computes how many miles are in $x$ feet. Find the algebraic rule for $M$. (If you do not know how many feet are in one mile, you can look it up on Google.)\n$M(x)=$ [ANS]\nEvaluate $M(25000)$ and interpret the result:\nThere are about [ANS] miles in [ANS] feet.", "answer_v1": [ "x/5280", "4.73485", "25000" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Suppose that $M$ is the function that computes how many miles are in $x$ feet. Find the algebraic rule for $M$. (If you do not know how many feet are in one mile, you can look it up on Google.)\n$M(x)=$ [ANS]\nEvaluate $M(11000)$ and interpret the result:\nThere are about [ANS] miles in [ANS] feet.", "answer_v2": [ "x/5280", "2.08333", "11000" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Suppose that $M$ is the function that computes how many miles are in $x$ feet. Find the algebraic rule for $M$. (If you do not know how many feet are in one mile, you can look it up on Google.)\n$M(x)=$ [ANS]\nEvaluate $M(16000)$ and interpret the result:\nThere are about [ANS] miles in [ANS] feet.", "answer_v3": [ "x/5280", "3.0303", "16000" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0182", "subject": "Algebra", "topic": "Functions", "subtopic": "Interpretation and applications", "level": "3", "keywords": [ "evaluate", "rational", "fraction", "multivariable" ], "problem_v1": "Target heart rate for moderate exercise is $50\\%$ to $70\\%$ of maximum heart rate. If we want to represent a certain percent of an individual\u2019s maximum heart rate, we\u2019d use the formula\n${\\text{rate}=p(220-a)}$ where $p$ is the percent, and $a$ is age in years. Determine the target heart rate at $65\\%$ level for someone who is $45$ years old. Round your answer to an integer. The target heart rate at $65\\%$ level for someone who is $45$ years old is [ANS] beats per minute.", "answer_v1": [ "114" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Target heart rate for moderate exercise is $50\\%$ to $70\\%$ of maximum heart rate. If we want to represent a certain percent of an individual\u2019s maximum heart rate, we\u2019d use the formula\n${\\text{rate}=p(220-a)}$ where $p$ is the percent, and $a$ is age in years. Determine the target heart rate at $52\\%$ level for someone who is $66$ years old. Round your answer to an integer. The target heart rate at $52\\%$ level for someone who is $66$ years old is [ANS] beats per minute.", "answer_v2": [ "80" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Target heart rate for moderate exercise is $50\\%$ to $70\\%$ of maximum heart rate. If we want to represent a certain percent of an individual\u2019s maximum heart rate, we\u2019d use the formula\n${\\text{rate}=p(220-a)}$ where $p$ is the percent, and $a$ is age in years. Determine the target heart rate at $56\\%$ level for someone who is $46$ years old. Round your answer to an integer. The target heart rate at $56\\%$ level for someone who is $46$ years old is [ANS] beats per minute.", "answer_v3": [ "97" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0183", "subject": "Algebra", "topic": "Functions", "subtopic": "Interpretation and applications", "level": "5", "keywords": [ "rational", "domain" ], "problem_v1": "The population of deer in a forest can be modeled by\n${P(x)={\\frac{2480x+2660}{8x+7}}}$ where $x$ is the number of years in the future. Answer the following questions. 1) How many deer live in this forest this year? [ANS]\n2) How many deer will live in this forest $12$ years later? Round your answer to an integer. [ANS]\n3) After how many years, the deer population will be $314$? Round your answer to an integer. [ANS]\n4) Use a calculator to answer this question: Many years in the future, about how many deer will live in this forest? [ANS]", "answer_v1": [ "380", "315", "13", "310" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "The population of deer in a forest can be modeled by\n${P(x)={\\frac{940x+600}{2x+3}}}$ where $x$ is the number of years in the future. Answer the following questions. 1) How many deer live in this forest this year? [ANS]\n2) How many deer will live in this forest $29$ years later? Round your answer to an integer. [ANS]\n3) After how many years, the deer population will be $442$? Round your answer to an integer. [ANS]\n4) Use a calculator to answer this question: Many years in the future, about how many deer will live in this forest? [ANS]", "answer_v2": [ "200", "457", "13", "470" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "The population of deer in a forest can be modeled by\n${P(x)={\\frac{1280x+600}{4x+3}}}$ where $x$ is the number of years in the future. Answer the following questions. 1) How many deer live in this forest this year? [ANS]\n2) How many deer will live in this forest $26$ years later? Round your answer to an integer. [ANS]\n3) After how many years, the deer population will be $317$? Round your answer to an integer. [ANS]\n4) Use a calculator to answer this question: Many years in the future, about how many deer will live in this forest? [ANS]", "answer_v3": [ "200", "317", "28", "320" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Algebra_0184", "subject": "Algebra", "topic": "Functions", "subtopic": "Interpretation and applications", "level": "2", "keywords": [ "functions", "definition of function", "function notation" ], "problem_v1": "Let $f(t)$ denote the number of people eating in a restaurant $t$ minutes after 5 PM. Answer the following questions: a) Which of the following statements best describes the significance of the expression $f(5)=17$? [ANS] A. Every 5 minutes, 17 more people are eating B. There are 5 people eating at 5:17 PM C. There are 17 people eating at 5:05 PM D. There are 17 people eating at 10:00 PM E. None of the above\nb) Which of the following statements best describes the significance of the expression $f(a)=34$? [ANS] A. Every 34 minutes, the number of people eating has increased by $a$ people B. $a$ hours after 5 PM there are 34 people eating C. At 5:34 PM there are $a$ people eating D. $a$ minutes after 5 PM there are 34 people eating E. None of the above\nc) Which of the following statements best describes the significance of the expression $f(34)=b$? [ANS] A. $b$ hours after 5 PM there are 34 people eating B. $b$ minutes after 5 PM there are 34 people eating C. Every 34 minutes, the number of people eating has increased by $b$ people D. At 5:34 PM there are $b$ people eating E. None of the above\nd) Which of the following statements best describes the significance of the expression $n=f(t)$? [ANS] A. Every $t$ minutes, $n$ more people have begun eating B. $t$ hours after 5 PM there are $n$ people eating C. $n$ minutes after 5 PM there are $t$ people eating D. $n$ hours after 5 PM there are $t$ people eating E. None of the above", "answer_v1": [ "C", "D", "D", "E" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Let $f(t)$ denote the number of people eating in a restaurant $t$ minutes after 5 PM. Answer the following questions: a) Which of the following statements best describes the significance of the expression $f(2)=21$? [ANS] A. Every 2 minutes, 21 more people are eating B. There are 2 people eating at 5:21 PM C. There are 21 people eating at 7:00 PM D. There are 21 people eating at 5:02 PM E. None of the above\nb) Which of the following statements best describes the significance of the expression $f(a)=26$? [ANS] A. $a$ minutes after 5 PM there are 26 people eating B. At 5:26 PM there are $a$ people eating C. Every 26 minutes, the number of people eating has increased by $a$ people D. $a$ hours after 5 PM there are 26 people eating E. None of the above\nc) Which of the following statements best describes the significance of the expression $f(26)=b$? [ANS] A. $b$ hours after 5 PM there are 26 people eating B. At 5:26 PM there are $b$ people eating C. Every 26 minutes, the number of people eating has increased by $b$ people D. $b$ minutes after 5 PM there are 26 people eating E. None of the above\nd) Which of the following statements best describes the significance of the expression $n=f(t)$? [ANS] A. $n$ hours after 5 PM there are $t$ people eating B. Every $t$ minutes, $n$ more people have begun eating C. $t$ hours after 5 PM there are $n$ people eating D. $n$ minutes after 5 PM there are $t$ people eating E. None of the above", "answer_v2": [ "D", "A", "B", "E" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Let $f(t)$ denote the number of people eating in a restaurant $t$ minutes after 5 PM. Answer the following questions: a) Which of the following statements best describes the significance of the expression $f(3)=17$? [ANS] A. Every 3 minutes, 17 more people are eating B. There are 17 people eating at 5:03 PM C. There are 17 people eating at 8:00 PM D. There are 3 people eating at 5:17 PM E. None of the above\nb) Which of the following statements best describes the significance of the expression $f(a)=32$? [ANS] A. $a$ minutes after 5 PM there are 32 people eating B. At 5:32 PM there are $a$ people eating C. $a$ hours after 5 PM there are 32 people eating D. Every 32 minutes, the number of people eating has increased by $a$ people E. None of the above\nc) Which of the following statements best describes the significance of the expression $f(32)=b$? [ANS] A. $b$ minutes after 5 PM there are 32 people eating B. At 5:32 PM there are $b$ people eating C. Every 32 minutes, the number of people eating has increased by $b$ people D. $b$ hours after 5 PM there are 32 people eating E. None of the above\nd) Which of the following statements best describes the significance of the expression $n=f(t)$? [ANS] A. $n$ minutes after 5 PM there are $t$ people eating B. Every $t$ minutes, $n$ more people have begun eating C. $n$ hours after 5 PM there are $t$ people eating D. $t$ hours after 5 PM there are $n$ people eating E. None of the above", "answer_v3": [ "B", "A", "B", "E" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Algebra_0185", "subject": "Algebra", "topic": "Functions", "subtopic": "Interpretation and applications", "level": "3", "keywords": [ "functions", "definition of function", "function notation" ], "problem_v1": "Since Roger Bannister broke the 4-minute mile on May 6, 1954, the record has been lowered by over sixteen seconds. The table below shows the year in which each new world record for the one-mile run was set and the times (in min:sec) of those records.\n$\\begin{array}{cccccccc}\\hline Year & Time & & Year & Time & & Year & Time \\\\ \\hline 1954 & 3:59.4 & & 1966 & 3:51.3 & & 1981 & 3:48.53 \\\\ \\hline 1954 & 3:58.0 & & 1967 & 3:51.1 & & 1981 & 3:48.40 \\\\ \\hline 1957 & 3:57.2 & & 1975 & 3:51.0 & & 1981 & 3:47.33 \\\\ \\hline 1958 & 3:54.5 & & 1975 & 3:49.4 & & 1985 & 3:46.32 \\\\ \\hline 1962 & 3:54.4 & & 1979 & 3:49.0 & & 1993 & 3:44.39 \\\\ \\hline 1964 & 3:54.1 & & 1980 & 3:48.8 & & 1999 & 3:43.13 \\\\ \\hline 1965 & 3:53.6 & & & & & & \\\\ \\hline \\end{array}$\na) Is the time a function of the year? [ANS] A. YES B. NO\nb) Is the year a function of the time? [ANS] A. YES B. NO\nLet $y(r)$ be the year in which the world record, $r$, was set. c) Which of the following statements best describes the meaning of the expression $y(\\mbox{3:47.33})=1981$? [ANS] A. A record of 3:47.33 was set in 1981 B. 1981 people ran the mile under a time of 3:47.33 C. The record was lowered by almost 4 seconds in 1981 D. Just after 3:47 PM the record was set in 1981 E. None of the above\nd) Evaluate $y(\\mbox{3:48.40})=$ [ANS]", "answer_v1": [ "B", "A", "A", "1981" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "NV" ], "options_v1": [ [ "A", "B" ], [ "A", "B" ], [ "A", "B", "C", "D", "E" ], [] ], "problem_v2": "Since Roger Bannister broke the 4-minute mile on May 6, 1954, the record has been lowered by over sixteen seconds. The table below shows the year in which each new world record for the one-mile run was set and the times (in min:sec) of those records.\n$\\begin{array}{cccccccc}\\hline Year & Time & & Year & Time & & Year & Time \\\\ \\hline 1954 & 3:59.4 & & 1966 & 3:51.3 & & 1981 & 3:48.53 \\\\ \\hline 1954 & 3:58.0 & & 1967 & 3:51.1 & & 1981 & 3:48.40 \\\\ \\hline 1957 & 3:57.2 & & 1975 & 3:51.0 & & 1981 & 3:47.33 \\\\ \\hline 1958 & 3:54.5 & & 1975 & 3:49.4 & & 1985 & 3:46.32 \\\\ \\hline 1962 & 3:54.4 & & 1979 & 3:49.0 & & 1993 & 3:44.39 \\\\ \\hline 1964 & 3:54.1 & & 1980 & 3:48.8 & & 1999 & 3:43.13 \\\\ \\hline 1965 & 3:53.6 & & & & & & \\\\ \\hline \\end{array}$\na) Is the time a function of the year? [ANS] A. YES B. NO\nb) Is the year a function of the time? [ANS] A. YES B. NO\nLet $y(r)$ be the year in which the world record, $r$, was set. c) Which of the following statements best describes the meaning of the expression $y(\\mbox{3:47.33})=1981$? [ANS] A. The record was lowered by almost 4 seconds in 1981 B. 1981 people ran the mile under a time of 3:47.33 C. Just after 3:47 PM the record was set in 1981 D. A record of 3:47.33 was set in 1981 E. None of the above\nd) Evaluate $y(\\mbox{3:58.0})=$ [ANS]", "answer_v2": [ "B", "A", "D", "1954" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "NV" ], "options_v2": [ [ "A", "B" ], [ "A", "B" ], [ "A", "B", "C", "D", "E" ], [] ], "problem_v3": "Since Roger Bannister broke the 4-minute mile on May 6, 1954, the record has been lowered by over sixteen seconds. The table below shows the year in which each new world record for the one-mile run was set and the times (in min:sec) of those records.\n$\\begin{array}{cccccccc}\\hline Year & Time & & Year & Time & & Year & Time \\\\ \\hline 1954 & 3:59.4 & & 1966 & 3:51.3 & & 1981 & 3:48.53 \\\\ \\hline 1954 & 3:58.0 & & 1967 & 3:51.1 & & 1981 & 3:48.40 \\\\ \\hline 1957 & 3:57.2 & & 1975 & 3:51.0 & & 1981 & 3:47.33 \\\\ \\hline 1958 & 3:54.5 & & 1975 & 3:49.4 & & 1985 & 3:46.32 \\\\ \\hline 1962 & 3:54.4 & & 1979 & 3:49.0 & & 1993 & 3:44.39 \\\\ \\hline 1964 & 3:54.1 & & 1980 & 3:48.8 & & 1999 & 3:43.13 \\\\ \\hline 1965 & 3:53.6 & & & & & & \\\\ \\hline \\end{array}$\na) Is the time a function of the year? [ANS] A. YES B. NO\nb) Is the year a function of the time? [ANS] A. YES B. NO\nLet $y(r)$ be the year in which the world record, $r$, was set. c) Which of the following statements best describes the meaning of the expression $y(\\mbox{3:47.33})=1981$? [ANS] A. A record of 3:47.33 was set in 1981 B. Just after 3:47 PM the record was set in 1981 C. The record was lowered by almost 4 seconds in 1981 D. 1981 people ran the mile under a time of 3:47.33 E. None of the above\nd) Evaluate $y(\\mbox{3:54.1})=$ [ANS]", "answer_v3": [ "B", "A", "A", "1964" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "NV" ], "options_v3": [ [ "A", "B" ], [ "A", "B" ], [ "A", "B", "C", "D", "E" ], [] ] }, { "id": "Algebra_0186", "subject": "Algebra", "topic": "Functions", "subtopic": "Interpretation and applications", "level": "2", "keywords": [ "functions", "definition of function", "function notation" ], "problem_v1": "The table below $A=f(d)$, the amount of money $A$ (in billions of dollars) in bills of denomination $d$ circulating in US currency in 2005. For example according to the table values below there were \\$60.2 billion worth of \\$50 bills in circulation.\n$\\begin{array}{ccccccc}\\hline Denomination (value of bill) & 1 & 5 & 10 & 20 & 50 & 100 \\\\ \\hline Dollar Value in Circulation & 8.4 & 9.7 & 14.8 & 110.1 & 60.2 & 524.5 \\\\ \\hline \\end{array}$\na) Find $f(50)=$ [ANS]\nb) Using your answer in (a), what was the total number of \\$50 bills (not amount of money) in circulation in 2005? There was a total number of [ANS] billion \\$50 bills in circulation. c) Are the following statements True or False? [ANS] 1. There were more 5 dollar bills than 20 dollar bills [ANS] 2. There were more 5 dollar bills than 10 dollar bills", "answer_v1": [ "60.2", "1.204", "F", "T" ], "answer_type_v1": [ "NV", "NV", "TF", "TF" ], "options_v1": [ [], [], [ "T", "F" ], [ "T", "F" ] ], "problem_v2": "The table below $A=f(d)$, the amount of money $A$ (in billions of dollars) in bills of denomination $d$ circulating in US currency in 2005. For example according to the table values below there were \\$60.2 billion worth of \\$50 bills in circulation.\n$\\begin{array}{ccccccc}\\hline Denomination (value of bill) & 1 & 5 & 10 & 20 & 50 & 100 \\\\ \\hline Dollar Value in Circulation & 8.4 & 9.7 & 14.8 & 110.1 & 60.2 & 524.5 \\\\ \\hline \\end{array}$\na) Find $f(5)=$ [ANS]\nb) Using your answer in (a), what was the total number of \\$5 bills (not amount of money) in circulation in 2005? There was a total number of [ANS] billion \\$5 bills in circulation. c) Are the following statements True or False? [ANS] 1. There were more 1 dollar bills than 5 dollar bills [ANS] 2. There were more 5 dollar bills than 10 dollar bills", "answer_v2": [ "9.7", "1.94", "T", "T" ], "answer_type_v2": [ "NV", "NV", "TF", "TF" ], "options_v2": [ [], [], [ "T", "F" ], [ "T", "F" ] ], "problem_v3": "The table below $A=f(d)$, the amount of money $A$ (in billions of dollars) in bills of denomination $d$ circulating in US currency in 2005. For example according to the table values below there were \\$60.2 billion worth of \\$50 bills in circulation.\n$\\begin{array}{ccccccc}\\hline Denomination (value of bill) & 1 & 5 & 10 & 20 & 50 & 100 \\\\ \\hline Dollar Value in Circulation & 8.4 & 9.7 & 14.8 & 110.1 & 60.2 & 524.5 \\\\ \\hline \\end{array}$\na) Find $f(10)=$ [ANS]\nb) Using your answer in (a), what was the total number of \\$10 bills (not amount of money) in circulation in 2005? There was a total number of [ANS] billion \\$10 bills in circulation. c) Are the following statements True or False? [ANS] 1. There were more 5 dollar bills than 10 dollar bills [ANS] 2. There were more 1 dollar bills than 5 dollar bills", "answer_v3": [ "14.8", "1.48", "T", "T" ], "answer_type_v3": [ "NV", "NV", "TF", "TF" ], "options_v3": [ [], [], [ "T", "F" ], [ "T", "F" ] ] }, { "id": "Algebra_0187", "subject": "Algebra", "topic": "Functions", "subtopic": "Interpretation and applications", "level": "3", "keywords": [ "transformations", "shifts", "translations", "stretch", "compression", "table" ], "problem_v1": "The table below gives the total cost, $C=f(n)$, for a carpenter to build $n$ wooden chairs.\n$\\begin{array}{ccccccc}\\hline n & 0 & 20 & 40 & 60 & 80 & 100 \\\\ \\hline C(n) & 5000 & 6000 & 6850 & 7500 & 7950 & 8250 \\\\ \\hline \\end{array}$\nEvaluate each of the expressions below:\n(a) $f(60)=$ [ANS]\n(b) $f(x)=$ [ANS] if $x=40$ (c) $z=$ [ANS] if $f(z)=6000$ (d) $f(0)=$ [ANS]\nFor each of the statements below decide which (if any) expression\n(a)-(d) above it correctly describes by selecting the appropriate letter in each pull-down menu. An expression may be described correctly by more than one statement, and some statements may not match any of the expressions. (e) The fixed costs of the carpenter. . (f) The cost of building 40 chairs. . (g) The total number of chairs that can be built at a cost of \\$40. . (h) The total number of chairs that can be built at a cost of \\$6000. . (i) The cost of building 60 chairs. . (j) The total number of chairs the carpenter must build in order to break even. .", "answer_v1": [ "7500", "6850", "20", "5000", "(d)", "(b)", "None of the above", "(c)", "(a)", "None of the above" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [], [], [], [], [ "(a)", "(b)", "(c)", "(d)", "None of the above" ], [ "(a)", "(b)", "(c)", "(d)", "None of the above" ], [ "(a)", "(b)", "(c)", "(d)", "None of the above" ], [ "(a)", "(b)", "(c)", "(d)", "None of the above" ], [ "(a)", "(b)", "(c)", "(d)", "None of the above" ], [ "(a)", "(b)", "(c)", "(d)", "None of the above" ] ], "problem_v2": "The table below gives the total cost, $C=f(n)$, for a carpenter to build $n$ wooden chairs.\n$\\begin{array}{ccccccc}\\hline n & 0 & 5 & 10 & 15 & 20 & 25 \\\\ \\hline C(n) & 6000 & 7000 & 7700 & 8300 & 8800 & 9100 \\\\ \\hline \\end{array}$\nEvaluate each of the expressions below:\n(a) $f(5)=$ [ANS]\n(b) $f(x)=$ [ANS] if $x=10$ (c) $z=$ [ANS] if $f(z)=8800$ (d) $f(0)=$ [ANS]\nFor each of the statements below decide which (if any) expression\n(a)-(d) above it correctly describes by selecting the appropriate letter in each pull-down menu. An expression may be described correctly by more than one statement, and some statements may not match any of the expressions. (e) The fixed costs of the carpenter. . (f) The total number of chairs that can be built at a cost of \\$8800. . (g) The total number of chairs the carpenter must build in order to break even. . (h) The cost of building 5 chairs. . (i) The cost of building 10 chairs. . (j) The total number of chairs that can be built at a cost of \\$5. .", "answer_v2": [ "7000", "7700", "20", "6000", "(d)", "(c)", "None of the above", "(a)", "(b)", "None of the above" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [], [], [], [], [ "(a)", "(b)", "(c)", "(d)", "None of the above" ], [ "(a)", "(b)", "(c)", "(d)", "None of the above" ], [ "(a)", "(b)", "(c)", "(d)", "None of the above" ], [ "(a)", "(b)", "(c)", "(d)", "None of the above" ], [ "(a)", "(b)", "(c)", "(d)", "None of the above" ], [ "(a)", "(b)", "(c)", "(d)", "None of the above" ] ], "problem_v3": "The table below gives the total cost, $C=f(n)$, for a carpenter to build $n$ wooden chairs.\n$\\begin{array}{ccccccc}\\hline n & 0 & 10 & 20 & 30 & 40 & 50 \\\\ \\hline C(n) & 5500 & 6500 & 7250 & 7850 & 8300 & 8650 \\\\ \\hline \\end{array}$\nEvaluate each of the expressions below:\n(a) $f(40)=$ [ANS]\n(b) $f(x)=$ [ANS] if $x=30$ (c) $z=$ [ANS] if $f(z)=7250$ (d) $f(0)=$ [ANS]\nFor each of the statements below decide which (if any) expression\n(a)-(d) above it correctly describes by selecting the appropriate letter in each pull-down menu. An expression may be described correctly by more than one statement, and some statements may not match any of the expressions. (e) The cost of building 30 chairs. . (f) The total number of chairs that can be built at a cost of \\$7250. . (g) The cost of building 40 chairs. . (h) The total number of chairs that can be built at a cost of \\$40. . (i) The total number of chairs the carpenter must build in order to break even. . (j) The fixed costs of the carpenter. .", "answer_v3": [ "8300", "7850", "20", "5500", "(b)", "(c)", "(a)", "None of the above", "None of the above", "(d)" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [], [], [], [], [ "(a)", "(b)", "(c)", "(d)", "None of the above" ], [ "(a)", "(b)", "(c)", "(d)", "None of the above" ], [ "(a)", "(b)", "(c)", "(d)", "None of the above" ], [ "(a)", "(b)", "(c)", "(d)", "None of the above" ], [ "(a)", "(b)", "(c)", "(d)", "None of the above" ], [ "(a)", "(b)", "(c)", "(d)", "None of the above" ] ] }, { "id": "Algebra_0188", "subject": "Algebra", "topic": "Functions", "subtopic": "Interpretation and applications", "level": "5", "keywords": [ "functions", "piecwise" ], "problem_v1": "Many printing presses are designed with large plates that print a fixed number of pages as a unit. Each unit is called a signature. A particular press prints signatures of 16 pages each. Suppose $C(p)$ is the cost of printing a book of $p$ pages, assuming each signature printed costs \\$0.16.\n(a) What is the cost of printing a book of 160 pages? \\$ [ANS]\nWhat if there are 161 pages? \\$ [ANS]\nWhat if there are $p$ pages? $C(p)=$ \\$ [ANS] where the cost is rounded (up/down) [ANS] for the next signature. (b) What are the domain and range of $C(p)$? Domain: all nonnegative integer multiples of (1 through 16) [ANS] Range: \u00a0 all nonnegative integer multiples of (0.01 through 0.20) [ANS] (c) On a piece of paper, sketch a graph of $C(p)$.", "answer_v1": [ "1.6", "1.76", "0.16*p/16", "up", "1", "0.16" ], "answer_type_v1": [ "NV", "NV", "EX", "MCS", "MCS", "MCS" ], "options_v1": [ [], [], [], [ "up", "down" ], [ "1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11", "12", "13", "14", "15", "16" ], [ "0.01", "0.02", "0.03", "0.04", "0.05", "0.06", "0.07", "0.08", "0.09", "0.10", "0.11", "0.12", "0.13", "0.14", "0.15", "0.16", "0.17", "0.18", "0.19", "0.20" ] ], "problem_v2": "Many printing presses are designed with large plates that print a fixed number of pages as a unit. Each unit is called a signature. A particular press prints signatures of 8 pages each. Suppose $C(p)$ is the cost of printing a book of $p$ pages, assuming each signature printed costs \\$0.20.\n(a) What is the cost of printing a book of 56 pages? \\$ [ANS]\nWhat if there are 57 pages? \\$ [ANS]\nWhat if there are $p$ pages? $C(p)=$ \\$ [ANS] where the cost is rounded (up/down) [ANS] for the next signature. (b) What are the domain and range of $C(p)$? Domain: all nonnegative integer multiples of (1 through 16) [ANS] Range: \u00a0 all nonnegative integer multiples of (0.01 through 0.20) [ANS] (c) On a piece of paper, sketch a graph of $C(p)$.", "answer_v2": [ "1.4", "1.6", "0.2*p/8", "up", "1", "0.20" ], "answer_type_v2": [ "NV", "NV", "EX", "MCS", "MCS", "MCS" ], "options_v2": [ [], [], [], [ "up", "down" ], [ "1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11", "12", "13", "14", "15", "16" ], [ "0.01", "0.02", "0.03", "0.04", "0.05", "0.06", "0.07", "0.08", "0.09", "0.10", "0.11", "0.12", "0.13", "0.14", "0.15", "0.16", "0.17", "0.18", "0.19", "0.20" ] ], "problem_v3": "Many printing presses are designed with large plates that print a fixed number of pages as a unit. Each unit is called a signature. A particular press prints signatures of 8 pages each. Suppose $C(p)$ is the cost of printing a book of $p$ pages, assuming each signature printed costs \\$0.16.\n(a) What is the cost of printing a book of 64 pages? \\$ [ANS]\nWhat if there are 65 pages? \\$ [ANS]\nWhat if there are $p$ pages? $C(p)=$ \\$ [ANS] where the cost is rounded (up/down) [ANS] for the next signature. (b) What are the domain and range of $C(p)$? Domain: all nonnegative integer multiples of (1 through 16) [ANS] Range: \u00a0 all nonnegative integer multiples of (0.01 through 0.20) [ANS] (c) On a piece of paper, sketch a graph of $C(p)$.", "answer_v3": [ "1.28", "1.44", "0.16*p/8", "up", "1", "0.16" ], "answer_type_v3": [ "NV", "NV", "EX", "MCS", "MCS", "MCS" ], "options_v3": [ [], [], [], [ "up", "down" ], [ "1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11", "12", "13", "14", "15", "16" ], [ "0.01", "0.02", "0.03", "0.04", "0.05", "0.06", "0.07", "0.08", "0.09", "0.10", "0.11", "0.12", "0.13", "0.14", "0.15", "0.16", "0.17", "0.18", "0.19", "0.20" ] ] }, { "id": "Algebra_0189", "subject": "Algebra", "topic": "Functions", "subtopic": "Interpretation and applications", "level": "3", "keywords": [ "functions", "evaluating", "solving", "input", "output" ], "problem_v1": "Chicago's average monthly rainfall, $R=f(t)$ inches, is given as a function of the month, $t$, where January is $t=1$, in the table below.\n$\\begin{array}{ccccccccc}\\hline t, month & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\\ \\hline R, inches & 1.8 & 1.8 & 2.7 & 3.1 & 3.5 & 3.7 & 3.5 & 3.4 \\\\ \\hline \\end{array}$\n(a) Solve $f(t)=3.4$. $t=$ [ANS]\nThe solution(s) to $f(t)=3.4$ can be interpreted as saying [ANS] A. Chicago's average rainfall increases by 3.4 inches in the month of May. B. Chicago's average rainfall is greatest in the month of May. C. Chicago's average rainfall in the month of August is 3.4 inches. D. Chicago's average rainfall is least in the month of August. E. None of the above\n(b) Solve $f(t)=f(5)$. $t=$ [ANS]\nThe solution(s) to $f(t)=f(5)$ can be interpreted as saying [ANS] A. Chicago's average rainfall is greatest in the month of May. B. Chicago's average rainfall is 3.5 inches in the month of May. C. Chicago's average rainfall is 3.5 inches in the month of July. D. Chicago's average rainfall is 3.5 inches in the months of May and July. E. None of the above", "answer_v1": [ "8", "C", "(5, 7)", "D" ], "answer_type_v1": [ "NV", "MCS", "UOL", "MCS" ], "options_v1": [ [], [ "A", "B", "C", "D", "E" ], [], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Chicago's average monthly rainfall, $R=f(t)$ inches, is given as a function of the month, $t$, where January is $t=1$, in the table below.\n$\\begin{array}{ccccccccc}\\hline t, month & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\\ \\hline R, inches & 1.8 & 1.8 & 2.7 & 3.1 & 3.5 & 3.7 & 3.5 & 3.4 \\\\ \\hline \\end{array}$\n(a) Solve $f(t)=2.7$. $t=$ [ANS]\nThe solution(s) to $f(t)=2.7$ can be interpreted as saying [ANS] A. Chicago's average rainfall increases by 2.7 inches in the month of August. B. Chicago's average rainfall is greatest in the month of August. C. Chicago's average rainfall is least in the month of March. D. Chicago's average rainfall in the month of March is 2.7 inches. E. None of the above\n(b) Solve $f(t)=f(1)$. $t=$ [ANS]\nThe solution(s) to $f(t)=f(1)$ can be interpreted as saying [ANS] A. Chicago's average rainfall is 1.8 inches in the months of January and February. B. Chicago's average rainfall is 1.8 inches in the month of February. C. Chicago's average rainfall is greatest in the month of January. D. Chicago's average rainfall is 1.8 inches in the month of January. E. None of the above", "answer_v2": [ "3", "D", "(1, 2)", "A" ], "answer_type_v2": [ "NV", "MCS", "UOL", "MCS" ], "options_v2": [ [], [ "A", "B", "C", "D", "E" ], [], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Chicago's average monthly rainfall, $R=f(t)$ inches, is given as a function of the month, $t$, where January is $t=1$, in the table below.\n$\\begin{array}{ccccccccc}\\hline t, month & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\\ \\hline R, inches & 1.8 & 1.8 & 2.7 & 3.1 & 3.5 & 3.7 & 3.5 & 3.4 \\\\ \\hline \\end{array}$\n(a) Solve $f(t)=3.1$. $t=$ [ANS]\nThe solution(s) to $f(t)=3.1$ can be interpreted as saying [ANS] A. Chicago's average rainfall increases by 3.1 inches in the month of May. B. Chicago's average rainfall in the month of April is 3.1 inches. C. Chicago's average rainfall is least in the month of April. D. Chicago's average rainfall is greatest in the month of May. E. None of the above\n(b) Solve $f(t)=f(5)$. $t=$ [ANS]\nThe solution(s) to $f(t)=f(5)$ can be interpreted as saying [ANS] A. Chicago's average rainfall is 3.5 inches in the months of May and July. B. Chicago's average rainfall is 3.5 inches in the month of July. C. Chicago's average rainfall is 3.5 inches in the month of May. D. Chicago's average rainfall is greatest in the month of May. E. None of the above", "answer_v3": [ "4", "B", "(5, 7)", "A" ], "answer_type_v3": [ "NV", "MCS", "UOL", "MCS" ], "options_v3": [ [], [ "A", "B", "C", "D", "E" ], [], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Algebra_0190", "subject": "Algebra", "topic": "Functions", "subtopic": "Interpretation and applications", "level": "5", "keywords": [ "functions", "evaluating", "solving", "input", "output" ], "problem_v1": "New York state income tax is based on taxable income which is part of a person's total income. The tax owed to the state is calculated using taxable income (not total income). In 2005, for a single person with a taxable income between \\$20,000 and \\$100,000, the tax owed was \\$973 plus 6.85\\% of the taxable income over \\$20,000. Answer the following questions, and DO NOT include any commas in your final answers.\n(a) Compute the tax owed by a person whose taxable income is \\$81,000. tax=\\$ [ANS] (round to nearest dollar) (b) Consider a lawyer whose taxable income is 90\\% of her total income, \\$ $x$, where $x$ is between \\$60,000 and \\$120,000. Write a formula for $T(x)$, the amount of taxable income (not the tax owed, yet). $T(x)=$ [ANS]\n(c) Write a formula for $L(x)$, the amount owed by the lawyer in part (b). $L(x)=$ [ANS]\n(d) Use $L(x)$ to evaluate the tax liability (amount owed) for $x=90,000$ and compare your results to part (a). $L(90000)=$ \\$ [ANS] (round to nearest dollar)", "answer_v1": [ "5152", "0.9*x", "(0.9*x-20000)*0.0685+973", "5152" ], "answer_type_v1": [ "NV", "EX", "EX", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "New York state income tax is based on taxable income which is part of a person's total income. The tax owed to the state is calculated using taxable income (not total income). In 2005, for a single person with a taxable income between \\$20,000 and \\$100,000, the tax owed was \\$973 plus 6.85\\% of the taxable income over \\$20,000. Answer the following questions, and DO NOT include any commas in your final answers.\n(a) Compute the tax owed by a person whose taxable income is \\$77,000. tax=\\$ [ANS] (round to nearest dollar) (b) Consider a lawyer whose taxable income is 70\\% of her total income, \\$ $x$, where $x$ is between \\$60,000 and \\$120,000. Write a formula for $T(x)$, the amount of taxable income (not the tax owed, yet). $T(x)=$ [ANS]\n(c) Write a formula for $L(x)$, the amount owed by the lawyer in part (b). $L(x)=$ [ANS]\n(d) Use $L(x)$ to evaluate the tax liability (amount owed) for $x=110,000$ and compare your results to part (a). $L(110000)=$ \\$ [ANS] (round to nearest dollar)", "answer_v2": [ "4878", "0.7*x", "(0.7*x-20000)*0.0685+973", "4878" ], "answer_type_v2": [ "NV", "EX", "EX", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "New York state income tax is based on taxable income which is part of a person's total income. The tax owed to the state is calculated using taxable income (not total income). In 2005, for a single person with a taxable income between \\$20,000 and \\$100,000, the tax owed was \\$973 plus 6.85\\% of the taxable income over \\$20,000. Answer the following questions, and DO NOT include any commas in your final answers.\n(a) Compute the tax owed by a person whose taxable income is \\$70,000. tax=\\$ [ANS] (round to nearest dollar) (b) Consider a lawyer whose taxable income is 70\\% of her total income, \\$ $x$, where $x$ is between \\$60,000 and \\$120,000. Write a formula for $T(x)$, the amount of taxable income (not the tax owed, yet). $T(x)=$ [ANS]\n(c) Write a formula for $L(x)$, the amount owed by the lawyer in part (b). $L(x)=$ [ANS]\n(d) Use $L(x)$ to evaluate the tax liability (amount owed) for $x=100,000$ and compare your results to part (a). $L(100000)=$ \\$ [ANS] (round to nearest dollar)", "answer_v3": [ "4398", "0.7*x", "(0.7*x-20000)*0.0685+973", "4398" ], "answer_type_v3": [ "NV", "EX", "EX", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Algebra_0191", "subject": "Algebra", "topic": "Functions", "subtopic": "Interpretation and applications", "level": "3", "keywords": [ "functions", "evaluating", "solving", "input", "output" ], "problem_v1": "A national park records data regarding the total fox population $F$ over a 12 month period, where $t=0$ means January 1, $t=1$ means February 1, and so on. Below is the table of values they recorded:\n$\\begin{array}{ccccccccccccc}\\hline t, month & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\\\ \\hline F, foxes & 150 & 143 & 125 & 100 & 75 & 57 & 50 & 57 & 75 & 100 & 125 & 143 \\\\ \\hline \\end{array}$\n(a) Is $F$ a function of $t$ [ANS] A. Yes B. No\n(b) Let $g(t)=F$ denote the fox population in month $t$. Find all solution(s) to the equation $g(t)=100$. If there is more than one solution, give your answer as a comma separated list of numbers. $t=$ [ANS]", "answer_v1": [ "A", "(3, 9)" ], "answer_type_v1": [ "MCS", "UOL" ], "options_v1": [ [ "A", "B" ], [] ], "problem_v2": "A national park records data regarding the total fox population $F$ over a 12 month period, where $t=0$ means January 1, $t=1$ means February 1, and so on. Below is the table of values they recorded:\n$\\begin{array}{ccccccccccccc}\\hline t, month & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\\\ \\hline F, foxes & 150 & 143 & 125 & 100 & 75 & 57 & 50 & 57 & 75 & 100 & 125 & 143 \\\\ \\hline \\end{array}$\n(a) Is $F$ a function of $t$ [ANS] A. Yes B. No\n(b) Let $g(t)=F$ denote the fox population in month $t$. Find all solution(s) to the equation $g(t)=143$. If there is more than one solution, give your answer as a comma separated list of numbers. $t=$ [ANS]", "answer_v2": [ "A", "(1, 11)" ], "answer_type_v2": [ "MCS", "UOL" ], "options_v2": [ [ "A", "B" ], [] ], "problem_v3": "A national park records data regarding the total fox population $F$ over a 12 month period, where $t=0$ means January 1, $t=1$ means February 1, and so on. Below is the table of values they recorded:\n$\\begin{array}{ccccccccccccc}\\hline t, month & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\\\ \\hline F, foxes & 150 & 143 & 125 & 100 & 75 & 57 & 50 & 57 & 75 & 100 & 125 & 143 \\\\ \\hline \\end{array}$\n(a) Is $F$ a function of $t$ [ANS] A. Yes B. No\n(b) Let $g(t)=F$ denote the fox population in month $t$. Find all solution(s) to the equation $g(t)=75$. If there is more than one solution, give your answer as a comma separated list of numbers. $t=$ [ANS]", "answer_v3": [ "A", "(4, 8)" ], "answer_type_v3": [ "MCS", "UOL" ], "options_v3": [ [ "A", "B" ], [] ] }, { "id": "Algebra_0192", "subject": "Algebra", "topic": "Functions", "subtopic": "Interpretation and applications", "level": "2", "keywords": [ "functions", "domain", "range", "input", "output", "interval notation" ], "problem_v1": "According to the 1993 World Almanac, the number of calories a person walking at 3 mph burns per minute depends on the person's weight as in the following table.\nCalories per minute as a function of weight\n$\\begin{array}{ccccccc}\\hline Weight (pounds) & 100 & 120 & 150 & 170 & 200 & 220 \\\\ \\hline Walking (calories) & 2.7 & 3.2 & 4 & 4.6 & 5.4 & 5.9 \\\\ \\hline \\end{array}$\n(a) On a sheet of paper, graph the linear function for the calories used per minute while walking, $C$, as a function of weight, $w$. Find an approximate formula for this equation. $C=$ [ANS]\n(b) What is a meaningful domain for your function? [ANS] A. $100 \\leq w \\leq 220$ B. $-\\infty < w < \\infty$ C. $0 \\leq w$ D. $0 \\leq w \\leq 1000$\n(c) What is a meaningful range for your function? [ANS] A. $0 \\leq C \\leq 6$ B. $0 \\leq C \\leq 100$ C. $-\\infty < C < \\infty$ D. $0 \\leq C$\n(d) Use your function from part (a) to determine how many calories per minute a person who weighs 185 pounds uses per minute of walking. [ANS]", "answer_v1": [ "-0.017+0.027*w", "A", "A", "5" ], "answer_type_v1": [ "EX", "MCS", "MCS", "NV" ], "options_v1": [ [], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [] ], "problem_v2": "According to the 1993 World Almanac, the number of calories a person walking at 3 mph burns per minute depends on the person's weight as in the following table.\nCalories per minute as a function of weight\n$\\begin{array}{ccccccc}\\hline Weight (pounds) & 100 & 120 & 150 & 170 & 200 & 220 \\\\ \\hline Walking (calories) & 2.7 & 3.2 & 4 & 4.6 & 5.4 & 5.9 \\\\ \\hline \\end{array}$\n(a) On a sheet of paper, graph the linear function for the calories used per minute while walking, $C$, as a function of weight, $w$. Find an approximate formula for this equation. $C=$ [ANS]\n(b) What is a meaningful domain for your function? [ANS] A. $100 \\leq w \\leq 220$ B. $-\\infty < w < \\infty$ C. $0 \\leq w$ D. $0 \\leq w \\leq 1000$\n(c) What is a meaningful range for your function? [ANS] A. $-\\infty < C < \\infty$ B. $0 \\leq C \\leq 100$ C. $0 \\leq C$ D. $0 \\leq C \\leq 6$\n(d) Use your function from part (a) to determine how many calories per minute a person who weighs 115 pounds uses per minute of walking. [ANS]", "answer_v2": [ "-0.017+0.027*w", "A", "D", "3.1" ], "answer_type_v2": [ "EX", "MCS", "MCS", "NV" ], "options_v2": [ [], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [] ], "problem_v3": "According to the 1993 World Almanac, the number of calories a person walking at 3 mph burns per minute depends on the person's weight as in the following table.\nCalories per minute as a function of weight\n$\\begin{array}{ccccccc}\\hline Weight (pounds) & 100 & 120 & 150 & 170 & 200 & 220 \\\\ \\hline Walking (calories) & 2.7 & 3.2 & 4 & 4.6 & 5.4 & 5.9 \\\\ \\hline \\end{array}$\n(a) On a sheet of paper, graph the linear function for the calories used per minute while walking, $C$, as a function of weight, $w$. Find an approximate formula for this equation. $C=$ [ANS]\n(b) What is a meaningful domain for your function? [ANS] A. $100 \\leq w \\leq 220$ B. $0 \\leq w$ C. $-\\infty < w < \\infty$ D. $0 \\leq w \\leq 1000$\n(c) What is a meaningful range for your function? [ANS] A. $-\\infty < C < \\infty$ B. $0 \\leq C \\leq 6$ C. $0 \\leq C \\leq 100$ D. $0 \\leq C$\n(d) Use your function from part (a) to determine how many calories per minute a person who weighs 135 pounds uses per minute of walking. [ANS]", "answer_v3": [ "-0.017+0.027*w", "A", "B", "3.6" ], "answer_type_v3": [ "EX", "MCS", "MCS", "NV" ], "options_v3": [ [], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [] ] }, { "id": "Algebra_0193", "subject": "Algebra", "topic": "Functions", "subtopic": "Interpretation and applications", "level": "5", "keywords": [ "composition", "decomposition", "function" ], "problem_v1": "Let $A=f(r)$ be the area of a circle with radius $r$ and $r=h(t)$ be the radius of the circle at time $t$. Which of the following statements correctly provides a practical interpretation of the composite function $f(h(t))$? Select all that apply if more than one is appropriate. [ANS] A. The length of the radius at time $t$. B. At what time $t$ the area will be $A=f(r)$. C. The area of the circle at time $t$. D. The area of the circle which at time $t$ has radius $h(t)$. E. At what time $t$ the radius will be $r=h(t)$. F. The length of the radius of a circle with area $A=f(r)$ at time $t$. G. None of the above", "answer_v1": [ "CD" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v2": "Let $A=f(r)$ be the area of a circle with radius $r$ and $r=h(t)$ be the radius of the circle at time $t$. Which of the following statements correctly provides a practical interpretation of the composite function $f(h(t))$? Select all that apply if more than one is appropriate. [ANS] A. At what time $t$ the area will be $A=f(r)$. B. At what time $t$ the radius will be $r=h(t)$. C. The length of the radius at time $t$. D. The area of the circle which at time $t$ has radius $h(t)$. E. The length of the radius of a circle with area $A=f(r)$ at time $t$. F. The area of the circle at time $t$. G. None of the above", "answer_v2": [ "DF" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v3": "Let $A=f(r)$ be the area of a circle with radius $r$ and $r=h(t)$ be the radius of the circle at time $t$. Which of the following statements correctly provides a practical interpretation of the composite function $f(h(t))$? Select all that apply if more than one is appropriate. [ANS] A. At what time $t$ the radius will be $r=h(t)$. B. At what time $t$ the area will be $A=f(r)$. C. The area of the circle which at time $t$ has radius $h(t)$. D. The length of the radius at time $t$. E. The area of the circle at time $t$. F. The length of the radius of a circle with area $A=f(r)$ at time $t$. G. None of the above", "answer_v3": [ "CE" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G" ] ] }, { "id": "Algebra_0194", "subject": "Algebra", "topic": "Functions", "subtopic": "Interpretation and applications", "level": "3", "keywords": [ "functions", "domain", "range", "input", "output", "interval notation" ], "problem_v1": "The area, $A=f(s)$ square feet, of a square wooden deck is a function of the side $s$ feet. Stain that costs \\$31.50 will cover 200 square feet of wood.\n(a) Write the formula for $f(s)$. $f(s)=$ [ANS]\n(b) Find a formula for $C=g(A)$, the cost in dollars of staining an area of $A$ square feet. $C=g(A)=$ [ANS]\n(c) Find $C=g(f(s))$. $C=g(f(s))=$ [ANS]\nWhich of the following best explains the meaning of the composite function $g(f(s))$? [ANS] A. The amount of stain to cover a square region of side length $s$ feet. B. The cost to stain a region of area $A$ square feet. C. The cost to stain a square region of side length $s$ feet. D. The function for square feet and cost of stain.\n(d) Evaluate and interpret, giving units. $f(9)=$ [ANS] [ANS] $g(90)=$ [ANS] [ANS] $g(f(11))=$ [ANS] [ANS]", "answer_v1": [ "s^2", "0.1575*A", "0.1575*s^2", "C", "81", "square feet", "14.175", "dollars", "19.0575", "dollars" ], "answer_type_v1": [ "EX", "EX", "EX", "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B", "C", "D" ], [], [ "feet", "square feet", "dollars", "dollars per foot", "dollars per square foot" ], [], [ "feet", "square feet", "dollars", "dollars per foot", "dollars per square foot" ], [], [ "feet", "square feet", "dollars", "dollars per foot", "dollars per square foot" ] ], "problem_v2": "The area, $A=f(s)$ square feet, of a square wooden deck is a function of the side $s$ feet. Stain that costs \\$27.50 will cover 220 square feet of wood.\n(a) Write the formula for $f(s)$. $f(s)=$ [ANS]\n(b) Find a formula for $C=g(A)$, the cost in dollars of staining an area of $A$ square feet. $C=g(A)=$ [ANS]\n(c) Find $C=g(f(s))$. $C=g(f(s))=$ [ANS]\nWhich of the following best explains the meaning of the composite function $g(f(s))$? [ANS] A. The amount of stain to cover a square region of side length $s$ feet. B. The cost to stain a region of area $A$ square feet. C. The function for square feet and cost of stain. D. The cost to stain a square region of side length $s$ feet.\n(d) Evaluate and interpret, giving units. $f(7)=$ [ANS] [ANS] $g(120)=$ [ANS] [ANS] $g(f(9))=$ [ANS] [ANS]", "answer_v2": [ "s^2", "0.125*A", "0.125*s^2", "D", "49", "square feet", "15", "dollars", "10.125", "dollars" ], "answer_type_v2": [ "EX", "EX", "EX", "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B", "C", "D" ], [], [ "feet", "square feet", "dollars", "dollars per foot", "dollars per square foot" ], [], [ "feet", "square feet", "dollars", "dollars per foot", "dollars per square foot" ], [], [ "feet", "square feet", "dollars", "dollars per foot", "dollars per square foot" ] ], "problem_v3": "The area, $A=f(s)$ square feet, of a square wooden deck is a function of the side $s$ feet. Stain that costs \\$28.50 will cover 200 square feet of wood.\n(a) Write the formula for $f(s)$. $f(s)=$ [ANS]\n(b) Find a formula for $C=g(A)$, the cost in dollars of staining an area of $A$ square feet. $C=g(A)=$ [ANS]\n(c) Find $C=g(f(s))$. $C=g(f(s))=$ [ANS]\nWhich of the following best explains the meaning of the composite function $g(f(s))$? [ANS] A. The amount of stain to cover a square region of side length $s$ feet. B. The cost to stain a square region of side length $s$ feet. C. The function for square feet and cost of stain. D. The cost to stain a region of area $A$ square feet.\n(d) Evaluate and interpret, giving units. $f(8)=$ [ANS] [ANS] $g(90)=$ [ANS] [ANS] $g(f(10))=$ [ANS] [ANS]", "answer_v3": [ "s^2", "0.1425*A", "0.1425*s^2", "B", "64", "square feet", "12.825", "dollars", "14.25", "dollars" ], "answer_type_v3": [ "EX", "EX", "EX", "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B", "C", "D" ], [], [ "feet", "square feet", "dollars", "dollars per foot", "dollars per square foot" ], [], [ "feet", "square feet", "dollars", "dollars per foot", "dollars per square foot" ], [], [ "feet", "square feet", "dollars", "dollars per foot", "dollars per square foot" ] ] }, { "id": "Algebra_0195", "subject": "Algebra", "topic": "Functions", "subtopic": "Interpretation and applications", "level": "5", "keywords": [ "domain", "function' 'composition", "algebra", "combining functions", "domain" ], "problem_v1": "The number of bacteria in a refrigerated food product is given by $N(T)=28 T^2-132 T+59$, $5 < T < 35$ where $T$ is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by $T(t)=4 t+1.3$, where $t$ is the time in hours. Find the composite function $N(T(t))$: $N(T(t))=$ [ANS]\nFind the time when the bacteria count reaches 21513 Time Needed=[ANS]", "answer_v1": [ "28*(4*t+1.3)**2 - 132*(4*t+1.3) + 59", "7.20947473332645" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The number of bacteria in a refrigerated food product is given by $N(T)=20 T^2-99 T+94$, $4 < T < 34$ where $T$ is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by $T(t)=9 t+1.3$, where $t$ is the time in hours. Find the composite function $N(T(t))$: $N(T(t))=$ [ANS]\nFind the time when the bacteria count reaches 6644 Time Needed=[ANS]", "answer_v2": [ "20*(9*t+1.3)**2 - 99*(9*t+1.3) + 94", "2.16004684071625" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The number of bacteria in a refrigerated food product is given by $N(T)=23 T^2-98 T+61$, $4 < T < 34$ where $T$ is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by $T(t)=3 t+1.3$, where $t$ is the time in hours. Find the composite function $N(T(t))$: $N(T(t))=$ [ANS]\nFind the time when the bacteria count reaches 12808 Time Needed=[ANS]", "answer_v3": [ "23*(3*t+1.3)**2 - 98*(3*t+1.3) + 61", "8.15615268105991" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0196", "subject": "Algebra", "topic": "Functions", "subtopic": "Interpretation and applications", "level": "5", "keywords": [ "domain", "function' 'composition' 'word problem", "algebra", "combining functions", "composition" ], "problem_v1": "A spherical weather balloon is being inflated. The radius of the balloon is increasing at the rate of 8 cm per second. Express the surface area of the balloon as a function of time $t$ (in seconds). If needed you can enter $\\pi$ as pi. Your answer is [ANS].", "answer_v1": [ "4*3.14159265358979*8*8*t*t" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "A spherical weather balloon is being inflated. The radius of the balloon is increasing at the rate of 2 cm per second. Express the surface area of the balloon as a function of time $t$ (in seconds). If needed you can enter $\\pi$ as pi. Your answer is [ANS].", "answer_v2": [ "4*3.14159265358979*2*2*t*t" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "A spherical weather balloon is being inflated. The radius of the balloon is increasing at the rate of 4 cm per second. Express the surface area of the balloon as a function of time $t$ (in seconds). If needed you can enter $\\pi$ as pi. Your answer is [ANS].", "answer_v3": [ "4*3.14159265358979*4*4*t*t" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0197", "subject": "Algebra", "topic": "Functions", "subtopic": "Interpretation and applications", "level": "2", "keywords": [ "Algebra", "Functions", "Domain" ], "problem_v1": "An open box is to be made from a flat piece of material 17 inches long and 4 inches wide by cutting equal squares of length $x$ from the corners and folding up the sides.\nWrite the volume $V$ of the box as a function of $x$. Leave it as a product of factors, do not multiply out the factors. $V(x)=$ [ANS]\nIf we write the domain of $V(x)$ as an open interval in the form $(a,b)$, then what is $a$? $a=$ [ANS]\nand what is $b$? $b=$ [ANS]", "answer_v1": [ "((17-2x)(4-2x)x)", "0", "2" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "An open box is to be made from a flat piece of material 8 inches long and 6 inches wide by cutting equal squares of length $x$ from the corners and folding up the sides.\nWrite the volume $V$ of the box as a function of $x$. Leave it as a product of factors, do not multiply out the factors. $V(x)=$ [ANS]\nIf we write the domain of $V(x)$ as an open interval in the form $(a,b)$, then what is $a$? $a=$ [ANS]\nand what is $b$? $b=$ [ANS]", "answer_v2": [ "((8-2x)(6-2x)x)", "0", "3" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "An open box is to be made from a flat piece of material 11 inches long and 5 inches wide by cutting equal squares of length $x$ from the corners and folding up the sides.\nWrite the volume $V$ of the box as a function of $x$. Leave it as a product of factors, do not multiply out the factors. $V(x)=$ [ANS]\nIf we write the domain of $V(x)$ as an open interval in the form $(a,b)$, then what is $a$? $a=$ [ANS]\nand what is $b$? $b=$ [ANS]", "answer_v3": [ "((11-2x)(5-2x)x)", "0", "2.5" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0198", "subject": "Algebra", "topic": "Functions", "subtopic": "Interpretation and applications", "level": "3", "keywords": [ "functions", "polynomials" ], "problem_v1": "The altitude of a right triangle is 16 cm. Let $h$ be the length of the hypotenuse and let $p$ be the perimeter of the triangle. Express $h$ as a function of $p$. $h(p)=$ [ANS]", "answer_v1": [ "(p**2 - 2 * 16 * p + 2 * 16**2)/(2 * (p - 16))" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "The altitude of a right triangle is 3 cm. Let $h$ be the length of the hypotenuse and let $p$ be the perimeter of the triangle. Express $h$ as a function of $p$. $h(p)=$ [ANS]", "answer_v2": [ "(p**2 - 2 * 3 * p + 2 * 3**2)/(2 * (p - 3))" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "The altitude of a right triangle is 7 cm. Let $h$ be the length of the hypotenuse and let $p$ be the perimeter of the triangle. Express $h$ as a function of $p$. $h(p)=$ [ANS]", "answer_v3": [ "(p**2 - 2 * 7 * p + 2 * 7**2)/(2 * (p - 7))" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0199", "subject": "Algebra", "topic": "Functions", "subtopic": "Interpretation and applications", "level": "5", "keywords": [ "average rate of change" ], "problem_v1": "Let $P_1$ and $P_2$ be the populations (in hundreds) of Town 1 and Town 2, respectively. The table below shows data for these two populations for five different years.\n$\\begin{array}{cccccc}\\hline Year & 1990 & 1993 & 1997 & 2002 & 2007 \\\\ \\hline P_1 & 49 & 57 & 65 & 73 & 81 \\\\ \\hline P_2 & 86 & 81 & 76 & 71 & 66 \\\\ \\hline \\end{array}$\nFind the average rate of change of each population over each of the time intervals below.\n(a) From 1990 to 1997, the average rate of change of the population of Town 1 was [ANS] hundred people per year, and the average rate of change of the population of Town 2 was [ANS] hundred people per year. (b) From 1997 to 2007, the average rate of change of the population of Town 1 was [ANS] hundred people per year, and the average rate of change of the population of Town 2 was [ANS] hundred people per year. (c) From 1990 to 2007, the average rate of change of the population of Town 1 was [ANS] hundred people per year, and the average rate of change of the population of Town 2 was [ANS] hundred people per year.", "answer_v1": [ "16/7", "-10/7", "16/10", "-10/10", "32/17", "-20/17" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Let $P_1$ and $P_2$ be the populations (in hundreds) of Town 1 and Town 2, respectively. The table below shows data for these two populations for five different years.\n$\\begin{array}{cccccc}\\hline Year & 1980 & 1984 & 1986 & 1990 & 1997 \\\\ \\hline P_1 & 48 & 52 & 56 & 60 & 64 \\\\ \\hline P_2 & 82 & 75 & 68 & 61 & 54 \\\\ \\hline \\end{array}$\nFind the average rate of change of each population over each of the time intervals below.\n(a) From 1980 to 1986, the average rate of change of the population of Town 1 was [ANS] hundred people per year, and the average rate of change of the population of Town 2 was [ANS] hundred people per year. (b) From 1986 to 1997, the average rate of change of the population of Town 1 was [ANS] hundred people per year, and the average rate of change of the population of Town 2 was [ANS] hundred people per year. (c) From 1980 to 1997, the average rate of change of the population of Town 1 was [ANS] hundred people per year, and the average rate of change of the population of Town 2 was [ANS] hundred people per year.", "answer_v2": [ "8/6", "-14/6", "8/11", "-14/11", "16/17", "-28/17" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Let $P_1$ and $P_2$ be the populations (in hundreds) of Town 1 and Town 2, respectively. The table below shows data for these two populations for five different years.\n$\\begin{array}{cccccc}\\hline Year & 1980 & 1983 & 1986 & 1991 & 1995 \\\\ \\hline P_1 & 49 & 59 & 69 & 79 & 89 \\\\ \\hline P_2 & 93 & 84 & 75 & 66 & 57 \\\\ \\hline \\end{array}$\nFind the average rate of change of each population over each of the time intervals below.\n(a) From 1980 to 1986, the average rate of change of the population of Town 1 was [ANS] hundred people per year, and the average rate of change of the population of Town 2 was [ANS] hundred people per year. (b) From 1986 to 1995, the average rate of change of the population of Town 1 was [ANS] hundred people per year, and the average rate of change of the population of Town 2 was [ANS] hundred people per year. (c) From 1980 to 1995, the average rate of change of the population of Town 1 was [ANS] hundred people per year, and the average rate of change of the population of Town 2 was [ANS] hundred people per year.", "answer_v3": [ "20/6", "-18/6", "20/9", "-18/9", "40/15", "-36/15" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Algebra_0200", "subject": "Algebra", "topic": "Functions", "subtopic": "Interpretation and applications", "level": "4", "keywords": [ "calculus", "composition of functions", "combining functions" ], "problem_v1": "Let $p$ be the price of an item and $q$ be the number of items sold at that price, where $q=f(p)$. What do the following quantities mean in terms of prices and quantities sold? 1. $f(45)$ is the [ANS] A. price for which 45 items are sold B. rate at which the price is changing when 45 items are sold. C. average price of the first 45 items sold. D. rate at which items are sold when the price is 45. E. revenue generated by the sale of 45 items. F. median number of items sold if the price is no more than 45. G. number of items sold when the price is 45.\n2. $f^{-1}(40)$ is the [ANS] A. average price of the first 40 items sold. B. price at which 40 items will be sold. C. rate at which items are sold when the price is 40. D. number of the first 40 items that are returned for refund. E. revenue generated by the sale of 40 items. F. rate at which the price is changing when 40 items are sold. G. number of items sold when the price is 40.", "answer_v1": [ "G", "B" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v2": "Let $p$ be the price of an item and $q$ be the number of items sold at that price, where $q=f(p)$. What do the following quantities mean in terms of prices and quantities sold? 1. $f(20)$ is the [ANS] A. average price of the first 20 items sold. B. number of items sold when the price is 20. C. revenue generated by the sale of 20 items. D. rate at which the price is changing when 20 items are sold. E. median number of items sold if the price is no more than 20. F. rate at which items are sold when the price is 20. G. price for which 20 items are sold\n2. $f^{-1}(50)$ is the [ANS] A. revenue generated by the sale of 50 items. B. rate at which the price is changing when 50 items are sold. C. rate at which items are sold when the price is 50. D. average price of the first 50 items sold. E. price at which 50 items will be sold. F. number of the first 50 items that are returned for refund. G. number of items sold when the price is 50.", "answer_v2": [ "B", "E" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v3": "Let $p$ be the price of an item and $q$ be the number of items sold at that price, where $q=f(p)$. What do the following quantities mean in terms of prices and quantities sold? 1. $f(30)$ is the [ANS] A. median number of items sold if the price is no more than 30. B. rate at which items are sold when the price is 30. C. price for which 30 items are sold D. average price of the first 30 items sold. E. number of items sold when the price is 30. F. rate at which the price is changing when 30 items are sold. G. revenue generated by the sale of 30 items.\n2. $f^{-1}(40)$ is the [ANS] A. rate at which items are sold when the price is 40. B. average price of the first 40 items sold. C. number of the first 40 items that are returned for refund. D. price at which 40 items will be sold. E. revenue generated by the sale of 40 items. F. rate at which the price is changing when 40 items are sold. G. number of items sold when the price is 40.", "answer_v3": [ "E", "D" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ] ] }, { "id": "Algebra_0201", "subject": "Algebra", "topic": "Transformations of functions and graphs", "subtopic": "Shifts: vertical and horizontal", "level": "3", "keywords": [ "transformations", "shifts", "translations", "graph" ], "problem_v1": "Describe a series of shifts which translates the graph $y=(x-7)^3-6$ back onto the graph of $y=x^3$. Select the correct direction using the pulldown menus, and enter a number which identifies the amount of units for each shift. We must shift the graph to the [ANS] by [ANS] units, and shift the graph [ANS] [ANS] units.", "answer_v1": [ "left", "7", "up", "6" ], "answer_type_v1": [ "MCS", "NV", "MCS", "NV" ], "options_v1": [ [ "left", "right" ], [], [ "up", "down" ], [] ], "problem_v2": "Describe a series of shifts which translates the graph $y=(x+3)^3-4$ back onto the graph of $y=x^3$. Select the correct direction using the pulldown menus, and enter a number which identifies the amount of units for each shift. We must shift the graph to the [ANS] by [ANS] units, and shift the graph [ANS] [ANS] units.", "answer_v2": [ "right", "3", "up", "4" ], "answer_type_v2": [ "MCS", "NV", "MCS", "NV" ], "options_v2": [ [ "left", "right" ], [], [ "up", "down" ], [] ], "problem_v3": "Describe a series of shifts which translates the graph $y=(x+5)^3-6$ back onto the graph of $y=x^3$. Select the correct direction using the pulldown menus, and enter a number which identifies the amount of units for each shift. We must shift the graph to the [ANS] by [ANS] units, and shift the graph [ANS] [ANS] units.", "answer_v3": [ "right", "5", "up", "6" ], "answer_type_v3": [ "MCS", "NV", "MCS", "NV" ], "options_v3": [ [ "left", "right" ], [], [ "up", "down" ], [] ] }, { "id": "Algebra_0202", "subject": "Algebra", "topic": "Transformations of functions and graphs", "subtopic": "Shifts: vertical and horizontal", "level": "2", "keywords": [ "transformations", "shifts", "translations", "graph" ], "problem_v1": "The graph of $g(x)$ contains the point $(-3,6)$. Write a formula for a translation of $g$ whose graph contains the point:\n(a) (-4, 6) $y=$ [ANS]\n(b) (-3, 10) $y=$ [ANS]\n(Note: Your answers should be an equation which is entered in function notation, meaning you should be entering a formula of the form $y=g(x-215)+45$.)", "answer_v1": [ "g(x+1)", "g(x)+4" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "The graph of $g(x)$ contains the point $(-9,8)$. Write a formula for a translation of $g$ whose graph contains the point:\n(a) (-4, 8) $y=$ [ANS]\n(b) (-9, 6) $y=$ [ANS]\n(Note: Your answers should be an equation which is entered in function notation, meaning you should be entering a formula of the form $y=g(x-215)+45$.)", "answer_v2": [ "g(x+-5)", "g(x)+-2" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "The graph of $g(x)$ contains the point $(-7,6)$. Write a formula for a translation of $g$ whose graph contains the point:\n(a) (-4, 6) $y=$ [ANS]\n(b) (-7, 2) $y=$ [ANS]\n(Note: Your answers should be an equation which is entered in function notation, meaning you should be entering a formula of the form $y=g(x-215)+45$.)", "answer_v3": [ "g(x+-3)", "g(x)+-4" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0203", "subject": "Algebra", "topic": "Transformations of functions and graphs", "subtopic": "Shifts: vertical and horizontal", "level": "2", "keywords": [ "transformations", "shifts", "translations", "graph" ], "problem_v1": "(a) The graph of $y=2^{x-3}$ is the graph of $y=2^{x}$ $\\begin{array}{ccc}\\hline shifted horizontally & & [ANS]right 3 unitsright 3 unitsright 4 unitsright 4 unitsright 5 unitsright 5 unitsand \\\\ \\hline shifted vertically & & [ANS]up 3 unitsup 3 unitsup 4 unitsup 4 unitsup 5 unitsup 5 units. \\\\ \\hline \\end{array}$\n(b) The graph of $y=e^{x}+1$ is the graph of $y=e^{x}$ $\\begin{array}{ccc}\\hline shifted horizontally & & [ANS]right 3 unitsright 3 unitsright 4 unitsright 4 unitsright 5 unitsright 5 unitsand \\\\ \\hline shifted vertically & & [ANS]up 3 unitsup 3 unitsup 4 unitsup 4 unitsup 5 unitsup 5 units. \\\\ \\hline \\end{array}$\n(c) The graph of $y=2^{x-1}+2$ is the graph of $y=2^{x}$ $\\begin{array}{ccc}\\hline shifted horizontally & & [ANS]right 3 unitsright 3 unitsright 4 unitsright 4 unitsright 5 unitsright 5 unitsand \\\\ \\hline shifted vertically & & [ANS]up 3 unitsup 3 unitsup 4 unitsup 4 unitsup 5 unitsup 5 units. \\\\ \\hline \\end{array}$\n(d) The graph of $y=e^{x+2}+9$ is the graph of $y=e^{x}+10$ $\\begin{array}{ccc}\\hline shifted horizontally & & [ANS]right 3 unitsright 3 unitsright 4 unitsright 4 unitsright 5 unitsright 5 unitsand \\\\ \\hline shifted vertically & & [ANS]up 3 unitsup 3 unitsup 4 unitsup 4 unitsup 5 unitsup 5 units. \\\\ \\hline \\end{array}$", "answer_v1": [ "right 3 units", "no vertical shift", "no horizontal shift", "up 1 unit", "right 1 unit", "up 2 units", "left 2 units", "down 1 unit" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "left 5 units", "left 4 units", "left 3 units", "left 2 units", "left 1 unit", "no horizontal shift", "right 1 unit", "right 2 units", "right 3 units", "right 4 units", "right 5 units" ], [ "down 5 units", "down 4 units", "down 3 units", "down 2 units", "down 1 unit", "no vertical shift", "up 1 unit", "up 2 units", "up 3 units", "up 4 units", "up 5 units" ], [ "left 5 units", "left 4 units", "left 3 units", "left 2 units", "left 1 unit", "no horizontal shift", "right 1 unit", "right 2 units", "right 3 units", "right 4 units", "right 5 units" ], [ "down 5 units", "down 4 units", "down 3 units", "down 2 units", "down 1 unit", "no vertical shift", "up 1 unit", "up 2 units", "up 3 units", "up 4 units", "up 5 units" ], [ "left 5 units", "left 4 units", "left 3 units", "left 2 units", "left 1 unit", "no horizontal shift", "right 1 unit", "right 2 units", "right 3 units", "right 4 units", "right 5 units" ], [ "down 5 units", "down 4 units", "down 3 units", "down 2 units", "down 1 unit", "no vertical shift", "up 1 unit", "up 2 units", "up 3 units", "up 4 units", "up 5 units" ], [ "left 5 units", "left 4 units", "left 3 units", "left 2 units", "left 1 unit", "no horizontal shift", "right 1 unit", "right 2 units", "right 3 units", "right 4 units", "right 5 units" ], [ "down 5 units", "down 4 units", "down 3 units", "down 2 units", "down 1 unit", "no vertical shift", "up 1 unit", "up 2 units", "up 3 units", "up 4 units", "up 5 units" ] ], "problem_v2": "(a) The graph of $y=2^{x+5}$ is the graph of $y=2^{x}$ $\\begin{array}{ccc}\\hline shifted horizontally & & [ANS]right 3 unitsright 3 unitsright 4 unitsright 4 unitsright 5 unitsright 5 unitsand \\\\ \\hline shifted vertically & & [ANS]up 3 unitsup 3 unitsup 4 unitsup 4 unitsup 5 unitsup 5 units. \\\\ \\hline \\end{array}$\n(b) The graph of $y=e^{x}+5$ is the graph of $y=e^{x}$ $\\begin{array}{ccc}\\hline shifted horizontally & & [ANS]right 3 unitsright 3 unitsright 4 unitsright 4 unitsright 5 unitsright 5 unitsand \\\\ \\hline shifted vertically & & [ANS]up 3 unitsup 3 unitsup 4 unitsup 4 unitsup 5 unitsup 5 units. \\\\ \\hline \\end{array}$\n(c) The graph of $y=2^{x+4}-2$ is the graph of $y=2^{x}$ $\\begin{array}{ccc}\\hline shifted horizontally & & [ANS]right 3 unitsright 3 unitsright 4 unitsright 4 unitsright 5 unitsright 5 unitsand \\\\ \\hline shifted vertically & & [ANS]up 3 unitsup 3 unitsup 4 unitsup 4 unitsup 5 unitsup 5 units. \\\\ \\hline \\end{array}$\n(d) The graph of $y=e^{x-5}+11$ is the graph of $y=e^{x}+10$ $\\begin{array}{ccc}\\hline shifted horizontally & & [ANS]right 3 unitsright 3 unitsright 4 unitsright 4 unitsright 5 unitsright 5 unitsand \\\\ \\hline shifted vertically & & [ANS]up 3 unitsup 3 unitsup 4 unitsup 4 unitsup 5 unitsup 5 units. \\\\ \\hline \\end{array}$", "answer_v2": [ "left 5 units", "no vertical shift", "no horizontal shift", "up 5 units", "left 4 units", "down 2 units", "right 5 units", "up 1 unit" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "left 5 units", "left 4 units", "left 3 units", "left 2 units", "left 1 unit", "no horizontal shift", "right 1 unit", "right 2 units", "right 3 units", "right 4 units", "right 5 units" ], [ "down 5 units", "down 4 units", "down 3 units", "down 2 units", "down 1 unit", "no vertical shift", "up 1 unit", "up 2 units", "up 3 units", "up 4 units", "up 5 units" ], [ "left 5 units", "left 4 units", "left 3 units", "left 2 units", "left 1 unit", "no horizontal shift", "right 1 unit", "right 2 units", "right 3 units", "right 4 units", "right 5 units" ], [ "down 5 units", "down 4 units", "down 3 units", "down 2 units", "down 1 unit", "no vertical shift", "up 1 unit", "up 2 units", "up 3 units", "up 4 units", "up 5 units" ], [ "left 5 units", "left 4 units", "left 3 units", "left 2 units", "left 1 unit", "no horizontal shift", "right 1 unit", "right 2 units", "right 3 units", "right 4 units", "right 5 units" ], [ "down 5 units", "down 4 units", "down 3 units", "down 2 units", "down 1 unit", "no vertical shift", "up 1 unit", "up 2 units", "up 3 units", "up 4 units", "up 5 units" ], [ "left 5 units", "left 4 units", "left 3 units", "left 2 units", "left 1 unit", "no horizontal shift", "right 1 unit", "right 2 units", "right 3 units", "right 4 units", "right 5 units" ], [ "down 5 units", "down 4 units", "down 3 units", "down 2 units", "down 1 unit", "no vertical shift", "up 1 unit", "up 2 units", "up 3 units", "up 4 units", "up 5 units" ] ], "problem_v3": "(a) The graph of $y=2^{x+2}$ is the graph of $y=2^{x}$ $\\begin{array}{ccc}\\hline shifted horizontally & & [ANS]right 3 unitsright 3 unitsright 4 unitsright 4 unitsright 5 unitsright 5 unitsand \\\\ \\hline shifted vertically & & [ANS]up 3 unitsup 3 unitsup 4 unitsup 4 unitsup 5 unitsup 5 units. \\\\ \\hline \\end{array}$\n(b) The graph of $y=e^{x}+1$ is the graph of $y=e^{x}$ $\\begin{array}{ccc}\\hline shifted horizontally & & [ANS]right 3 unitsright 3 unitsright 4 unitsright 4 unitsright 5 unitsright 5 unitsand \\\\ \\hline shifted vertically & & [ANS]up 3 unitsup 3 unitsup 4 unitsup 4 unitsup 5 unitsup 5 units. \\\\ \\hline \\end{array}$\n(c) The graph of $y=2^{x+2}+1$ is the graph of $y=2^{x}$ $\\begin{array}{ccc}\\hline shifted horizontally & & [ANS]right 3 unitsright 3 unitsright 4 unitsright 4 unitsright 5 unitsright 5 unitsand \\\\ \\hline shifted vertically & & [ANS]up 3 unitsup 3 unitsup 4 unitsup 4 unitsup 5 unitsup 5 units. \\\\ \\hline \\end{array}$\n(d) The graph of $y=e^{x-3}+8$ is the graph of $y=e^{x}+10$ $\\begin{array}{ccc}\\hline shifted horizontally & & [ANS]right 3 unitsright 3 unitsright 4 unitsright 4 unitsright 5 unitsright 5 unitsand \\\\ \\hline shifted vertically & & [ANS]up 3 unitsup 3 unitsup 4 unitsup 4 unitsup 5 unitsup 5 units. \\\\ \\hline \\end{array}$", "answer_v3": [ "left 2 units", "no vertical shift", "no horizontal shift", "up 1 unit", "left 2 units", "up 1 unit", "right 3 units", "down 2 units" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "left 5 units", "left 4 units", "left 3 units", "left 2 units", "left 1 unit", "no horizontal shift", "right 1 unit", "right 2 units", "right 3 units", "right 4 units", "right 5 units" ], [ "down 5 units", "down 4 units", "down 3 units", "down 2 units", "down 1 unit", "no vertical shift", "up 1 unit", "up 2 units", "up 3 units", "up 4 units", "up 5 units" ], [ "left 5 units", "left 4 units", "left 3 units", "left 2 units", "left 1 unit", "no horizontal shift", "right 1 unit", "right 2 units", "right 3 units", "right 4 units", "right 5 units" ], [ "down 5 units", "down 4 units", "down 3 units", "down 2 units", "down 1 unit", "no vertical shift", "up 1 unit", "up 2 units", "up 3 units", "up 4 units", "up 5 units" ], [ "left 5 units", "left 4 units", "left 3 units", "left 2 units", "left 1 unit", "no horizontal shift", "right 1 unit", "right 2 units", "right 3 units", "right 4 units", "right 5 units" ], [ "down 5 units", "down 4 units", "down 3 units", "down 2 units", "down 1 unit", "no vertical shift", "up 1 unit", "up 2 units", "up 3 units", "up 4 units", "up 5 units" ], [ "left 5 units", "left 4 units", "left 3 units", "left 2 units", "left 1 unit", "no horizontal shift", "right 1 unit", "right 2 units", "right 3 units", "right 4 units", "right 5 units" ], [ "down 5 units", "down 4 units", "down 3 units", "down 2 units", "down 1 unit", "no vertical shift", "up 1 unit", "up 2 units", "up 3 units", "up 4 units", "up 5 units" ] ] }, { "id": "Algebra_0205", "subject": "Algebra", "topic": "Transformations of functions and graphs", "subtopic": "Shifts: vertical and horizontal", "level": "2", "keywords": [ "algebra", "number", "theory", "transformation of function" ], "problem_v1": "Given $f(x)=x^2$, after performing the following transformations: shift upward 74 units and shift 59 units to the right, the new function $g(x)=$ [ANS]", "answer_v1": [ "(x-59)^2+74" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Given $f(x)=x^2$, after performing the following transformations: shift upward 10 units and shift 93 units to the right, the new function $g(x)=$ [ANS]", "answer_v2": [ "(x-93)^2+10" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Given $f(x)=x^2$, after performing the following transformations: shift upward 32 units and shift 61 units to the right, the new function $g(x)=$ [ANS]", "answer_v3": [ "(x-61)^2+32" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0206", "subject": "Algebra", "topic": "Transformations of functions and graphs", "subtopic": "Scale changes: vertical and horizontal", "level": "3", "keywords": [ "reflection", "compression", "expansion", "transformations" ], "problem_v1": "Describe the effect that the transformations $y=\\frac{1}{8} f \\left(\\frac{1}{8} x \\right)$ have on the graph of $y=f(x)$.\n(a) The graph of $y=f(x)$ is horizontally [ANS] by a factor of [ANS] and vertically [ANS] by a factor of [ANS]. Your numerical answers should all be positive. (b) If the point $(-2,-24)$ is on the graph of $y=f(x)$, what point must be on the graph of $y=\\frac{1}{8} f \\left(\\frac{1}{8} x \\right)$? The point [ANS] must be on the new graph.", "answer_v1": [ "stretched", "8", "compressed", "0.125", "(-16,-3)" ], "answer_type_v1": [ "MCS", "NV", "MCS", "NV", "OL" ], "options_v1": [ [ "stretched", "compressed" ], [], [ "stretched", "compressed" ], [], [] ], "problem_v2": "Describe the effect that the transformations $y=\\frac{1}{2} f \\left(\\frac{1}{2} x \\right)$ have on the graph of $y=f(x)$.\n(a) The graph of $y=f(x)$ is horizontally [ANS] by a factor of [ANS] and vertically [ANS] by a factor of [ANS]. Your numerical answers should all be positive. (b) If the point $(-2, 10)$ is on the graph of $y=f(x)$, what point must be on the graph of $y=\\frac{1}{2} f \\left(\\frac{1}{2} x \\right)$? The point [ANS] must be on the new graph.", "answer_v2": [ "stretched", "2", "compressed", "0.5", "(-4,5)" ], "answer_type_v2": [ "MCS", "NV", "MCS", "NV", "OL" ], "options_v2": [ [ "stretched", "compressed" ], [], [ "stretched", "compressed" ], [], [] ], "problem_v3": "Describe the effect that the transformations $y=\\frac{1}{4} f \\left(\\frac{1}{4} x \\right)$ have on the graph of $y=f(x)$.\n(a) The graph of $y=f(x)$ is horizontally [ANS] by a factor of [ANS] and vertically [ANS] by a factor of [ANS]. Your numerical answers should all be positive. (b) If the point $(3,-4)$ is on the graph of $y=f(x)$, what point must be on the graph of $y=\\frac{1}{4} f \\left(\\frac{1}{4} x \\right)$? The point [ANS] must be on the new graph.", "answer_v3": [ "stretched", "4", "compressed", "0.25", "(12,-1)" ], "answer_type_v3": [ "MCS", "NV", "MCS", "NV", "OL" ], "options_v3": [ [ "stretched", "compressed" ], [], [ "stretched", "compressed" ], [], [] ] }, { "id": "Algebra_0207", "subject": "Algebra", "topic": "Transformations of functions and graphs", "subtopic": "Shift and scale change", "level": "3", "keywords": [ "reflection", "compression", "expansion", "transformations" ], "problem_v1": "Every day I take the same taxi over the same route from home to the train station. The trip is $x$ miles, so the cost for the trip is $f(x)$. Match each story in (a)-(d) to a function in (i)-(iv) representing the amount paid to the driver. A. \u00a0 The meter in the taxi went crazy and showed five times the number of miles I actually traveled. B. \u00a0 I had a new driver today and he got lost. He drove five extra miles and charged me for it. C. \u00a0 I haven't paid my driver all week. Today is Friday and I'll pay what I owe for the week. D. \u00a0 I received a raise yesterday, so today I gave my driver a five dollar tip. (i) $f(x+5)$ matches statement [ANS] (ii) $f(x)+5$ matches statement [ANS] (iii) $f(5x)$ matches statement [ANS] (iv) $5f(x)$ matches statement [ANS]", "answer_v1": [ "B", "D", "A", "C" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Every day I take the same taxi over the same route from home to the train station. The trip is $x$ miles, so the cost for the trip is $f(x)$. Match each story in (a)-(d) to a function in (i)-(iv) representing the amount paid to the driver. A. \u00a0 I received a raise yesterday, so today I gave my driver a five dollar tip. B. \u00a0 The meter in the taxi went crazy and showed five times the number of miles I actually traveled. C. \u00a0 I had a new driver today and he got lost. He drove five extra miles and charged me for it. D. \u00a0 I haven't paid my driver all week. Today is Friday and I'll pay what I owe for the week. (i) $f(5x)$ matches statement [ANS] (ii) $f(x)+5$ matches statement [ANS] (iii) $f(x+5)$ matches statement [ANS] (iv) $5f(x)$ matches statement [ANS]", "answer_v2": [ "B", "A", "C", "D" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Every day I take the same taxi over the same route from home to the train station. The trip is $x$ miles, so the cost for the trip is $f(x)$. Match each story in (a)-(d) to a function in (i)-(iv) representing the amount paid to the driver. A. \u00a0 I had a new driver today and he got lost. He drove five extra miles and charged me for it. B. \u00a0 I haven't paid my driver all week. Today is Friday and I'll pay what I owe for the week. C. \u00a0 I received a raise yesterday, so today I gave my driver a five dollar tip. D. \u00a0 The meter in the taxi went crazy and showed five times the number of miles I actually traveled. (i) $f(x)+5$ matches statement [ANS] (ii) $5f(x)$ matches statement [ANS] (iii) $f(5x)$ matches statement [ANS] (iv) $f(x+5)$ matches statement [ANS]", "answer_v3": [ "C", "B", "D", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Algebra_0208", "subject": "Algebra", "topic": "Transformations of functions and graphs", "subtopic": "Shift and scale change", "level": "3", "keywords": [ "reflection", "compression", "expansion", "transformations" ], "problem_v1": "The US population in millions is $P(t)$ today, where time $t$ is measured in years. For each statement, choose one of the expressions A-H that represents it.\n$\\begin{array}{cccc}\\hline & [ANS]GGHH 1. Today's population plus 10 million immigrants. [ANS]GGHH 2. Ten percent of the population we have today. [ANS]GGHH 3. The population after 100,000 people have emigrated. [ANS]GGHH 4. The population 10 years before today. & & A. P(t+10) B. P(t)+10 C. P(t)-0.1 D. P(t)+0.1 E. 0.1 P(t) F. P(t)/0.1 G. P(t-10) H. P(t)-10 \\\\ \\hline \\end{array}$", "answer_v1": [ "B", "E", "C", "G" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ] ], "problem_v2": "The US population in millions is $P(t)$ today, where time $t$ is measured in years. For each statement, choose one of the expressions A-H that represents it.\n$\\begin{array}{cccc}\\hline & [ANS]GGHH 1. Ten percent of the population we have today. [ANS]GGHH 2. The population after 100,000 people have emigrated. [ANS]GGHH 3. Today's population plus 10 million immigrants. [ANS]GGHH 4. The population 10 years before today. & & A. P(t)+0.1 B. P(t+10) C. P(t)+10 D. P(t-10) E. P(t)-0.1 F. 0.1 P(t) G. P(t)/0.1 H. P(t)-10 \\\\ \\hline \\end{array}$", "answer_v2": [ "F", "E", "C", "D" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ] ], "problem_v3": "The US population in millions is $P(t)$ today, where time $t$ is measured in years. For each statement, choose one of the expressions A-H that represents it.\n$\\begin{array}{cccc}\\hline & [ANS]GGHH 1. Ten percent of the population we have today. [ANS]GGHH 2. The population 10 years before today. [ANS]GGHH 3. Today's population plus 10 million immigrants. [ANS]GGHH 4. The population after 100,000 people have emigrated. & & A. 0.1 P(t) B. P(t-10) C. P(t)+0.1 D. P(t)-10 E. P(t+10) F. P(t)-0.1 G. P(t)+10 H. P(t)/0.1 \\\\ \\hline \\end{array}$", "answer_v3": [ "A", "B", "G", "F" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ] ] }, { "id": "Algebra_0209", "subject": "Algebra", "topic": "Transformations of functions and graphs", "subtopic": "Shift and scale change", "level": "3", "keywords": [ "transformations", "transformations of functions", "translation", "shifts", "stretch", "logarithms" ], "problem_v1": "Applying a horizontal stretch by a factor of $p$ (where $p$ is a constant such that $p >1$) to $h (x)=\\ln x$ is equivalent to applying what shift to $h$? Give both the direction and the amount of the shift (in terms of $p$). Answer: This is equivalent to applying a shift [ANS] by [ANS] units.", "answer_v1": [ "Down", "ln(p)" ], "answer_type_v1": [ "MCS", "EX" ], "options_v1": [ [ "Left", "Right", "Up", "Down" ], [] ], "problem_v2": "Applying a horizontal stretch by a factor of $c$ (where $c$ is a constant such that $c >1$) to $w (x)=\\ln x$ is equivalent to applying what shift to $w$? Give both the direction and the amount of the shift (in terms of $c$). Answer: This is equivalent to applying a shift [ANS] by [ANS] units.", "answer_v2": [ "Down", "ln(c)" ], "answer_type_v2": [ "MCS", "EX" ], "options_v2": [ [ "Left", "Right", "Up", "Down" ], [] ], "problem_v3": "Applying a horizontal stretch by a factor of $k$ (where $k$ is a constant such that $k >1$) to $h (x)=\\ln x$ is equivalent to applying what shift to $h$? Give both the direction and the amount of the shift (in terms of $k$). Answer: This is equivalent to applying a shift [ANS] by [ANS] units.", "answer_v3": [ "Down", "ln(k)" ], "answer_type_v3": [ "MCS", "EX" ], "options_v3": [ [ "Left", "Right", "Up", "Down" ], [] ] }, { "id": "Algebra_0210", "subject": "Algebra", "topic": "Transformations of functions and graphs", "subtopic": "Symmetry: even, odd, neither", "level": "2", "keywords": [ "rational", "fraction", "numerator", "denominator", "asymptote" ], "problem_v1": "Determine whether each of the following rational functions is even, odd, or neither.\n[ANS] 1. $ \\frac{x^3+4x^2}{7x^2+1}$\n[ANS] 2. $ \\frac{4x^3+2x}{7x^4}$\n[ANS] 3. $ \\frac{4x^3+2x}{7x^5+3x^3}$\n[ANS] 4. $ \\frac{2x^4+8x^3}{x^3+4x^2}$\n[ANS] 5. $ \\frac{x^2-4}{x-2}$\n[ANS] 6. $ \\frac{1}{x^2+9x^6}$\n[ANS] 7. $ \\frac{4x^2+7}{7x^8+3x^2}$\n[ANS] 8. $ \\frac{x^3}{7x^2+1}$", "answer_v1": [ "neither", "odd", "even", "odd", "neither", "even", "even", "odd" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "even", "odd", "neither" ], [ "even", "odd", "neither" ], [ "even", "odd", "neither" ], [ "even", "odd", "neither" ], [ "even", "odd", "neither" ], [ "even", "odd", "neither" ], [ "even", "odd", "neither" ], [ "even", "odd", "neither" ] ], "problem_v2": "Determine whether each of the following rational functions is even, odd, or neither.\n[ANS] 1. $ \\frac{x^3+4x^2}{7x^2+1}$\n[ANS] 2. $ \\frac{4x^3+2x}{7x^4}$\n[ANS] 3. $ \\frac{1}{x^2+9x^6}$\n[ANS] 4. $ \\frac{x^3}{7x^2+1}$\n[ANS] 5. $ \\frac{2x^3+8x^2}{x^3+4x^2}$\n[ANS] 6. $ \\frac{2x^4+8x^3}{x^3+4x^2}$\n[ANS] 7. $ \\frac{x^2-4}{x-2}$\n[ANS] 8. $ \\frac{4x^3+2x}{7x^5+3x^3}$", "answer_v2": [ "neither", "odd", "even", "odd", "even", "odd", "neither", "even" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "even", "odd", "neither" ], [ "even", "odd", "neither" ], [ "even", "odd", "neither" ], [ "even", "odd", "neither" ], [ "even", "odd", "neither" ], [ "even", "odd", "neither" ], [ "even", "odd", "neither" ], [ "even", "odd", "neither" ] ], "problem_v3": "Determine whether each of the following rational functions is even, odd, or neither.\n[ANS] 1. $ \\frac{4x^3+2x}{7x^4}$\n[ANS] 2. $ \\frac{1}{x^2+9x^6}$\n[ANS] 3. $ \\frac{1}{x^2}-\\frac{5}{x^8}$\n[ANS] 4. $ \\frac{4x^3+2x}{7x^5+3x^3}$\n[ANS] 5. $ \\frac{x^2-4}{x-2}$\n[ANS] 6. $ \\frac{2x^3+8x^2}{x^3+4x^2}$\n[ANS] 7. $ \\frac{4x^2+7}{7x^8+3x^2}$\n[ANS] 8. $ \\frac{x^3+4x^2}{7x^2+1}$", "answer_v3": [ "odd", "even", "even", "even", "neither", "even", "even", "neither" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "even", "odd", "neither" ], [ "even", "odd", "neither" ], [ "even", "odd", "neither" ], [ "even", "odd", "neither" ], [ "even", "odd", "neither" ], [ "even", "odd", "neither" ], [ "even", "odd", "neither" ], [ "even", "odd", "neither" ] ] }, { "id": "Algebra_0211", "subject": "Algebra", "topic": "Transformations of functions and graphs", "subtopic": "Symmetry: even, odd, neither", "level": "2", "keywords": [ "rational", "fraction", "numerator", "denominator", "asymptote" ], "problem_v1": "How does the symmetry of $ f(x)=\\frac{p(x)}{q(x)}$ depend on the symmetry of $p(x)$ and $q(x)$? Select all true statements. There may be more than one correct answer. [ANS] A. If $p(x)$ is even and $q(x)$ is odd, then $f(x)$ is even. B. If $p(x)$ and $q(x)$ are both even, then $f(x)$ is even. C. If $p(x)$ is even and $q(x)$ is odd, then $f(x)$ is odd. D. If $p(x)$ or $q(x)$ is neither even nor odd, then $f(x)$ is neither even nor odd. E. If $p(x)$ and $q(x)$ are both odd, then $f(x)$ is even. F. If $p(x)$ is odd and $q(x)$ is even, then $f(x)$ is even. G. If $p(x)$ is odd and $q(x)$ is even, then $f(x)$ is odd. H. If $p(x)$ and $q(x)$ are both odd, then $f(x)$ is odd.", "answer_v1": [ "BCEG" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G", "H" ] ], "problem_v2": "How does the symmetry of $ f(x)=\\frac{p(x)}{q(x)}$ depend on the symmetry of $p(x)$ and $q(x)$? Select all true statements. There may be more than one correct answer. [ANS] A. If $p(x)$ or $q(x)$ is neither even nor odd, then $f(x)$ is neither even nor odd. B. If $p(x)$ is even and $q(x)$ is odd, then $f(x)$ is even. C. If $p(x)$ is even and $q(x)$ is odd, then $f(x)$ is odd. D. If $p(x)$ is odd and $q(x)$ is even, then $f(x)$ is odd. E. If $p(x)$ and $q(x)$ are both odd, then $f(x)$ is even. F. If $p(x)$ and $q(x)$ are both even, then $f(x)$ is even. G. If $p(x)$ is odd and $q(x)$ is even, then $f(x)$ is even. H. If $p(x)$ and $q(x)$ are both odd, then $f(x)$ is odd.", "answer_v2": [ "CDEF" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G", "H" ] ], "problem_v3": "How does the symmetry of $ f(x)=\\frac{p(x)}{q(x)}$ depend on the symmetry of $p(x)$ and $q(x)$? Select all true statements. There may be more than one correct answer. [ANS] A. If $p(x)$ and $q(x)$ are both even, then $f(x)$ is even. B. If $p(x)$ and $q(x)$ are both odd, then $f(x)$ is even. C. If $p(x)$ or $q(x)$ is neither even nor odd, then $f(x)$ is neither even nor odd. D. If $p(x)$ and $q(x)$ are both odd, then $f(x)$ is odd. E. If $p(x)$ is even and $q(x)$ is odd, then $f(x)$ is even. F. If $p(x)$ is odd and $q(x)$ is even, then $f(x)$ is odd. G. If $p(x)$ is even and $q(x)$ is odd, then $f(x)$ is odd. H. If $p(x)$ is odd and $q(x)$ is even, then $f(x)$ is even.", "answer_v3": [ "ABFG" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G", "H" ] ] }, { "id": "Algebra_0212", "subject": "Algebra", "topic": "Transformations of functions and graphs", "subtopic": "Symmetry: even, odd, neither", "level": "3", "keywords": [ "algebra", "calculus" ], "problem_v1": "Enter a T or an F in each answer space below to indicate whether the corresponding statement is true or false. You must get all of the answers correct to receive credit. [ANS] 1. A function cannot be both even and odd. [ANS] 2. The composition of an odd function and an odd function is even [ANS] 3. The composition of an even and an odd function is even [ANS] 4. The sum of an even and an odd function is usually neither even or odd, but it may be even. [ANS] 5. The sum of two even functions is even [ANS] 6. The product of two even function is even [ANS] 7. The product of two odd function is odd [ANS] 8. The ratio of two odd functions is odd\nAll of the answers must be correct before you get credit for the problem.", "answer_v1": [ "F", "F", "T", "T", "T", "T", "F", "F" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Enter a T or an F in each answer space below to indicate whether the corresponding statement is true or false. You must get all of the answers correct to receive credit. [ANS] 1. The product of two even function is even [ANS] 2. The product of two odd function is odd [ANS] 3. The sum of two even functions is even [ANS] 4. The composition of an even and an odd function is even [ANS] 5. A function cannot be both even and odd. [ANS] 6. The ratio of two odd functions is odd [ANS] 7. The composition of an odd function and an odd function is even [ANS] 8. The sum of an even and an odd function is usually neither even or odd, but it may be even.\nAll of the answers must be correct before you get credit for the problem.", "answer_v2": [ "T", "F", "T", "T", "F", "F", "F", "T" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Enter a T or an F in each answer space below to indicate whether the corresponding statement is true or false. You must get all of the answers correct to receive credit. [ANS] 1. The ratio of two odd functions is odd [ANS] 2. The sum of an even and an odd function is usually neither even or odd, but it may be even. [ANS] 3. The sum of two even functions is even [ANS] 4. The composition of an odd function and an odd function is even [ANS] 5. The product of two even function is even [ANS] 6. A function cannot be both even and odd. [ANS] 7. The product of two odd function is odd [ANS] 8. The composition of an even and an odd function is even\nAll of the answers must be correct before you get credit for the problem.", "answer_v3": [ "F", "T", "T", "F", "T", "F", "F", "T" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Algebra_0213", "subject": "Algebra", "topic": "Transformations of functions and graphs", "subtopic": "Symmetry: even, odd, neither", "level": "3", "keywords": [ "algebra", "piecewise", "function" ], "problem_v1": "Determine the symmetries (if any) of the graphs of the given relations.\n(a) $7 y=3x^2-4$: [ANS] (b) $xy=8$: [ANS]", "answer_v1": [ "y-axis", "origin" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "x-axis", "y-axis", "origin", "All of the above", "None of the above" ], [ "x-axis", "y-axis", "origin", "All of the above", "None of the above" ] ], "problem_v2": "Determine the symmetries (if any) of the graphs of the given relations.\n(a) $y^2=x-10$: [ANS] (b) $10 y=3x^2-2$: [ANS]", "answer_v2": [ "x-axis", "y-axis" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "x-axis", "y-axis", "origin", "All of the above", "None of the above" ], [ "x-axis", "y-axis", "origin", "All of the above", "None of the above" ] ], "problem_v3": "Determine the symmetries (if any) of the graphs of the given relations.\n(a) $3 y=7x^2-6x$: [ANS] (b) $y^2=x-3$: [ANS]", "answer_v3": [ "None of the above", "x-axis" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "x-axis", "y-axis", "origin", "All of the above", "None of the above" ], [ "x-axis", "y-axis", "origin", "All of the above", "None of the above" ] ] }, { "id": "Algebra_0214", "subject": "Algebra", "topic": "Transformations of functions and graphs", "subtopic": "Three or more transformations", "level": "3", "keywords": [ "transformations", "transformations of functions", "translation", "shifts", "stretch", "compression", "composition", "order of operations" ], "problem_v1": "The graph of a function $y=n (c)$ has a horizontal asymptote at $y=2$ and a vertical asymptote at $c=5$. Find equations for the asymptotes of $y=3-0.3 n(-4 c)$. Answers: $\\quad y=$ [ANS] $\\quad$ and $\\quad c=$ [ANS]", "answer_v1": [ "2.4", "-1.25" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The graph of a function $y=v (d)$ has a horizontal asymptote at $y=-8$ and a vertical asymptote at $d=-3$. Find equations for the asymptotes of $y=8-0.3 v(-2 d)$. Answers: $\\quad y=$ [ANS] $\\quad$ and $\\quad d=$ [ANS]", "answer_v2": [ "10.4", "1.5" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The graph of a function $y=w (c)$ has a horizontal asymptote at $y=-4$ and a vertical asymptote at $c=3$. Find equations for the asymptotes of $y=2-0.3 w(-6 c)$. Answers: $\\quad y=$ [ANS] $\\quad$ and $\\quad c=$ [ANS]", "answer_v3": [ "3.2", "-0.5" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0215", "subject": "Algebra", "topic": "Transformations of functions and graphs", "subtopic": "Vertical shifts", "level": "3", "keywords": [ "reflection", "compression", "expansion", "transformations" ], "problem_v1": "The table below gives values of $\\ T=f(d)$, the average temperature (in degrees Celsius) at a depth d meters in a borehole in Belleterre, Quebec.\n$\\begin{array}{ccccccccc}\\hline d, depth (m) & 25 & 40 & 75 & 100 & 125 & 150 & 175 & 200 \\\\ \\hline T, temp (C) & 5.5 & 5.2 & 5.1 & 5.1 & 5.3 & 5.5 & 5.75 & 6 \\\\ \\hline \\end{array}$\nConsider the function $\\ g(d)=f(d)+3$ which describes another borehole near Belleterre.\n(a) Fill in all of the blanks in the table of values for $\\ g(d)$ for which you have sufficient information. If are unable to determine a value in the table, enter NONE. Do not leave any blanks in the table.\n$\\begin{array}{ccccccccc}\\hline d & 25 & 40 & 75 & 100 & 125 & 150 & 175 & 200 \\\\ \\hline g(d) & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\n(b) Which of the statements below best describes in words what the function $g(d)$ tells you about the borehole? [ANS] A. If the temperatures in both boreholes are the same, then you will be 3 meters closer to the top of this borehole than if you were in the borehole in Belleterre. B. Temperatures in this borehole are 3 degrees Celsius cooler than at the same depth in the Belleterre borehole. C. Temperatures in this borehole are 3 degrees Celsius warmer than at the same depth in the Belleterre borehole. D. If the temperatures in both boreholes are the same, then you will be 3 meters deeper in this borehole than if you were in the borehole in Belleterre. E. None of the above", "answer_v1": [ "8.5", "8.2", "8.1", "8.1", "8.3", "8.5", "8.75", "9", "C" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [], [], [], [], [], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "The table below gives values of $\\ T=f(d)$, the average temperature (in degrees Celsius) at a depth d meters in a borehole in Belleterre, Quebec.\n$\\begin{array}{ccccccccc}\\hline d, depth (m) & 25 & 40 & 75 & 100 & 125 & 150 & 175 & 200 \\\\ \\hline T, temp (C) & 5.5 & 5.2 & 5.1 & 5.1 & 5.3 & 5.5 & 5.75 & 6 \\\\ \\hline \\end{array}$\nConsider the function $\\ g(d)=f(d)-4$ which describes another borehole near Belleterre.\n(a) Fill in all of the blanks in the table of values for $\\ g(d)$ for which you have sufficient information. If are unable to determine a value in the table, enter NONE. Do not leave any blanks in the table.\n$\\begin{array}{ccccccccc}\\hline d & 25 & 40 & 75 & 100 & 125 & 150 & 175 & 200 \\\\ \\hline g(d) & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\n(b) Which of the statements below best describes in words what the function $g(d)$ tells you about the borehole? [ANS] A. If the temperatures in both boreholes are the same, then you will be 4 meters closer to the top of this borehole than if you were in the borehole in Belleterre. B. Temperatures in this borehole are 4 degrees Celsius warmer than at the same depth in the Belleterre borehole. C. If the temperatures in both boreholes are the same, then you will be 4 meters deeper in this borehole than if you were in the borehole in Belleterre. D. Temperatures in this borehole are 4 degrees Celsius cooler than at the same depth in the Belleterre borehole. E. None of the above", "answer_v2": [ "1.5", "1.2", "1.1", "1.1", "1.3", "1.5", "1.75", "2", "D" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [], [], [], [], [], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "The table below gives values of $\\ T=f(d)$, the average temperature (in degrees Celsius) at a depth d meters in a borehole in Belleterre, Quebec.\n$\\begin{array}{ccccccccc}\\hline d, depth (m) & 25 & 40 & 75 & 100 & 125 & 150 & 175 & 200 \\\\ \\hline T, temp (C) & 5.5 & 5.2 & 5.1 & 5.1 & 5.3 & 5.5 & 5.75 & 6 \\\\ \\hline \\end{array}$\nConsider the function $\\ g(d)=f(d)-3$ which describes another borehole near Belleterre.\n(a) Fill in all of the blanks in the table of values for $\\ g(d)$ for which you have sufficient information. If are unable to determine a value in the table, enter NONE. Do not leave any blanks in the table.\n$\\begin{array}{ccccccccc}\\hline d & 25 & 40 & 75 & 100 & 125 & 150 & 175 & 200 \\\\ \\hline g(d) & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\n(b) Which of the statements below best describes in words what the function $g(d)$ tells you about the borehole? [ANS] A. If the temperatures in both boreholes are the same, then you will be 3 meters closer to the top of this borehole than if you were in the borehole in Belleterre. B. Temperatures in this borehole are 3 degrees Celsius cooler than at the same depth in the Belleterre borehole. C. If the temperatures in both boreholes are the same, then you will be 3 meters deeper in this borehole than if you were in the borehole in Belleterre. D. Temperatures in this borehole are 3 degrees Celsius warmer than at the same depth in the Belleterre borehole. E. None of the above", "answer_v3": [ "2.5", "2.2", "2.1", "2.1", "2.3", "2.5", "2.75", "3", "B" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [], [], [], [], [], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Algebra_0216", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Finding the slope", "level": "2", "keywords": [ "line", "slope", "slope formula" ], "problem_v1": "A line passes through the points $(4,-6)$ and $(9,-16)$. Find this line\u2019s slope. If the slope does not exists, you may enter DNE or NONE.\nThis line\u2019s slope is [ANS].", "answer_v1": [ "-2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A line passes through the points $(1,4)$ and $(7,-26)$. Find this line\u2019s slope. If the slope does not exists, you may enter DNE or NONE.\nThis line\u2019s slope is [ANS].", "answer_v2": [ "-5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A line passes through the points $(2,-6)$ and $(8,-30)$. Find this line\u2019s slope. If the slope does not exists, you may enter DNE or NONE.\nThis line\u2019s slope is [ANS].", "answer_v3": [ "-4" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0217", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Finding the slope", "level": "2", "keywords": [ "Algebra", "Lines" ], "problem_v1": "Find the slope of the line passing through the points $(a,5 a+2)$ and $(a+h,5(a+2 h)+2)$.\nThe slope is [ANS]", "answer_v1": [ "10" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the slope of the line passing through the points $(a,-8 a+8)$ and $(a+h,-8(a-7 h)+8)$.\nThe slope is [ANS]", "answer_v2": [ "56" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the slope of the line passing through the points $(a,-4 a+2)$ and $(a+h,-4(a-4 h)+2)$.\nThe slope is [ANS]", "answer_v3": [ "16" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0218", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Finding the slope", "level": "2", "keywords": [ "algebra", "lines" ], "problem_v1": "The equation of the line that goes through the points $(-2,-5)$ and $(7,8)$ can be written in the form $y=mx+b$ where its slope $m$ is: [ANS]", "answer_v1": [ "1.44444444444444" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The equation of the line that goes through the points $(-5,-1)$ and $(2,4)$ can be written in the form $y=mx+b$ where its slope $m$ is: [ANS]", "answer_v2": [ "0.714285714285714" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The equation of the line that goes through the points $(-4,-4)$ and $(3,6)$ can be written in the form $y=mx+b$ where its slope $m$ is: [ANS]", "answer_v3": [ "1.42857142857143" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0219", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Parallel and perpendicular lines", "level": "3", "keywords": [], "problem_v1": "Find the value of $k$ so that the line containing the points $(-3,k)$ and $(1,-2)$ is perpendicular to the line $y=\\frac{5}{6} x+2$. $k=$ [ANS]", "answer_v1": [ "14/5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the value of $k$ so that the line containing the points $(-3,-3)$ and $(-5,k)$ is perpendicular to the line $y=-\\frac{1}{9} x-2$. $k=$ [ANS]", "answer_v2": [ "-21" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the value of $k$ so that the line containing the points $(6,-4)$ and $(-5,k)$ is perpendicular to the line $y=-\\frac{2}{3} x+3$. $k=$ [ANS]", "answer_v3": [ "-41/2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0220", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Equations of lines: slope-intercept form", "level": "3", "keywords": [ "linear expressions" ], "problem_v1": "(a) Put the expression $5x k+4 k+6-11x$ into the form $b+m x$. [ANS] $+\\Big($ [ANS] $\\Big) x$.\n(b) Is the expression $5xy+4x+6-11 y$ linear in the variable $x$? If it is linear, enter the slope. If it is not linear, enter NO. NO. [ANS]", "answer_v1": [ "4*k+6", "5*k-11", "5*y-11" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "(a) Put the expression $2x k+5 k+3-10x$ into the form $b+m x$. [ANS] $+\\Big($ [ANS] $\\Big) x$.\n(b) Is the expression $2xy+5x+3-10 y$ linear in the variable $x$? If it is linear, enter the slope. If it is not linear, enter NO. NO. [ANS]", "answer_v2": [ "5*k+3", "2*k-10", "2*y-10" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "(a) Put the expression $3x k+4 k+4-11x$ into the form $b+m x$. [ANS] $+\\Big($ [ANS] $\\Big) x$.\n(b) Is the expression $3xy+4x+4-11 y$ linear in the variable $x$? If it is linear, enter the slope. If it is not linear, enter NO. NO. [ANS]", "answer_v3": [ "4*k+4", "3*k-11", "3*y-11" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0221", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Equations of lines: general", "level": "3", "keywords": [ "coordinates", "linear equation" ], "problem_v1": "Write a linear equation which fits data in the table.\n$\\begin{array}{cc}\\hline x-values & y-values \\\\ \\hline-1 & {0} \\\\ \\hline 0 & {3} \\\\ \\hline 1 & {6} \\\\ \\hline 2 & {9} \\\\ \\hline 3 & {12} \\\\ \\hline \\end{array}$\nThe linear equation is $y=$ [ANS].", "answer_v1": [ "3*x+3" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Write a linear equation which fits data in the table.\n$\\begin{array}{cc}\\hline x-values & y-values \\\\ \\hline-4 & {-23} \\\\ \\hline-3 & {-19} \\\\ \\hline-2 & {-15} \\\\ \\hline-1 & {-11} \\\\ \\hline 0 & {-7} \\\\ \\hline \\end{array}$\nThe linear equation is $y=$ [ANS].", "answer_v2": [ "4*x-7" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Write a linear equation which fits data in the table.\n$\\begin{array}{cc}\\hline x-values & y-values \\\\ \\hline-3 & {-14} \\\\ \\hline-2 & {-11} \\\\ \\hline-1 & {-8} \\\\ \\hline 0 & {-5} \\\\ \\hline 1 & {-2} \\\\ \\hline \\end{array}$\nThe linear equation is $y=$ [ANS].", "answer_v3": [ "3*x-5" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0222", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Linear functions", "level": "2", "keywords": [ "lines", "linear functions", "slope-intercept", "point-slope form" ], "problem_v1": "Find an equation for the linear function which has $f(250)=1600$ and $f(650)=4200$ $f(x)=$ [ANS]", "answer_v1": [ "6.5*x+-25" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find an equation for the linear function which has $f(50)=1100$ and $f(800)=3800$ $f(x)=$ [ANS]", "answer_v2": [ "3.6*x+920" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find an equation for the linear function which has $f(100)=1300$ and $f(600)=4100$ $f(x)=$ [ANS]", "answer_v3": [ "5.6*x+740" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0223", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Linear functions", "level": "3", "keywords": [ "rate of change", "lines", "linear functions", "slope", "intercept" ], "problem_v1": "Decide whether each of the tables below could represent a linear function or not. If so enter Yes and if not enter No in the blank space below each table. a) $\\begin{array}{ccccc}\\hline x & 3 & 6 & 9 & 12 \\\\ \\hline h(x) & 15 & 10 & 0 &-10 \\\\ \\hline \\end{array}$\nCould this be a linear function? [ANS] b) $\\begin{array}{ccccc}\\hline x & 2 & 17 & 32 & 47 \\\\ \\hline f(x) & 42 & 30 & 18 & 6 \\\\ \\hline \\end{array}$\nCould this be a linear function? [ANS] c) $\\begin{array}{ccccc}\\hline x & 6 & 14 & 22 & 30 \\\\ \\hline g(x) &-16 &-28 &-42 &-56 \\\\ \\hline \\end{array}$\nCould this be a linear function? [ANS]", "answer_v1": [ "no", "yes", "no" ], "answer_type_v1": [ "TF", "TF", "TF" ], "options_v1": [ [ "yes", "no" ], [ "yes", "no" ], [ "yes", "no" ] ], "problem_v2": "Decide whether each of the tables below could represent a linear function or not. If so enter Yes and if not enter No in the blank space below each table. a) $\\begin{array}{ccccc}\\hline x &-3 & 2 & 7 & 12 \\\\ \\hline h(x) & 15 & 25 & 45 & 65 \\\\ \\hline \\end{array}$\nCould this be a linear function? [ANS] b) $\\begin{array}{ccccc}\\hline x & 4 & 14 & 24 & 34 \\\\ \\hline f(x) & 34 & 26 & 18 & 10 \\\\ \\hline \\end{array}$\nCould this be a linear function? [ANS] c) $\\begin{array}{ccccc}\\hline x & 0 & 2 & 4 & 6 \\\\ \\hline g(x) & 40 & 48 & 56 & 64 \\\\ \\hline \\end{array}$\nCould this be a linear function? [ANS]", "answer_v2": [ "no", "yes", "yes" ], "answer_type_v2": [ "TF", "TF", "TF" ], "options_v2": [ [ "yes", "no" ], [ "yes", "no" ], [ "yes", "no" ] ], "problem_v3": "Decide whether each of the tables below could represent a linear function or not. If so enter Yes and if not enter No in the blank space below each table. a) $\\begin{array}{ccccc}\\hline x &-6 & 3 & 12 & 21 \\\\ \\hline h(x) &-15 &-17 &-21 &-25 \\\\ \\hline \\end{array}$\nCould this be a linear function? [ANS] b) $\\begin{array}{ccccc}\\hline x &-1 & 2 & 5 & 8 \\\\ \\hline f(x) & 25 & 15 &-5 &-25 \\\\ \\hline \\end{array}$\nCould this be a linear function? [ANS] c) $\\begin{array}{ccccc}\\hline x & 0 & 4 & 8 & 12 \\\\ \\hline g(x) & 60 & 68 & 76 & 84 \\\\ \\hline \\end{array}$\nCould this be a linear function? [ANS]", "answer_v3": [ "no", "no", "yes" ], "answer_type_v3": [ "TF", "TF", "TF" ], "options_v3": [ [ "yes", "no" ], [ "yes", "no" ], [ "yes", "no" ] ] }, { "id": "Algebra_0224", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Linear functions", "level": "3", "keywords": [ "algebra", "slope", "line", "linear", "change", "fraction" ], "problem_v1": "y is a linear function of x with slope $-4$. If y is $2$ when x is $5$, what is y when x is $10$? [ANS]\nYou do NOT need to find the linear relationship between x and y to answer this question.", "answer_v1": [ "-18" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "y is a linear function of x with slope $-9$. If y is $9$ when x is $-9$, what is y when x is $-12$? [ANS]\nYou do NOT need to find the linear relationship between x and y to answer this question.", "answer_v2": [ "36" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "y is a linear function of x with slope $-8$. If y is $2$ when x is $-4$, what is y when x is $-3$? [ANS]\nYou do NOT need to find the linear relationship between x and y to answer this question.", "answer_v3": [ "-6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0226", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Linear functions", "level": "3", "keywords": [ "linear", "equations" ], "problem_v1": "Solve the following equation for $w$: 8 w+6 k+m=p. $w=$ [ANS]", "answer_v1": [ "(p-6*k-m)/8" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Solve the following equation for $w$: 2 w+9 k+m=p. $w=$ [ANS]", "answer_v2": [ "(p-9*k-m)/2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Solve the following equation for $w$: 4 w+6 k+m=p. $w=$ [ANS]", "answer_v3": [ "(p-6*k-m)/4" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0227", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Linear equations", "level": "1", "keywords": [ "coordinate", "cartesian", "line", "linear", "equation", "fraction" ], "problem_v1": "Consider the equation\n$y=-\\frac{5}{8}x$ Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer. [ANS] A. \\((-32,22)\\) B. \\((0,0)\\) C. \\((-24,15)\\) D. \\((32,-16)\\)", "answer_v1": [ "BC" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "Consider the equation\n$y=-\\frac{3}{4}x$ Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer. [ANS] A. \\((-20,15)\\) B. \\((4,-1)\\) C. \\((0,0)\\) D. \\((-4,5)\\)", "answer_v2": [ "AC" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "Consider the equation\n$y=-\\frac{3}{4}x$ Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer. [ANS] A. \\((-4,3)\\) B. \\((0,0)\\) C. \\((-16,14)\\) D. \\((8,-3)\\)", "answer_v3": [ "AB" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Algebra_0228", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Linear equations", "level": "4", "keywords": [ "true-false" ], "problem_v1": "Indicate with T (true) if the equations below are equivalent to a linear equation and with an F (False) if not. For example, the equation x^2+3x+4=x^2+5 turns into the linear equation 3x+4=5 after subtracting $x^2$ on both sides. Actually, the issue of whether a given equation is equivalent to a linear equation is quite subtle, but for the problems given here the answer will be (hopefully) obvious. To help you along, in this problem WeBWorK will indicate for each answer separately whether it's right or wrong and so you can get credit by just trying T's and F's. Of course you should think about the questions carefully and figure out why the answer is T or F. This will enable you to approach other questions more effectively. [ANS] 1. $\\frac{x}{6}-\\frac{2}{3}=\\frac{5}{6}$ [ANS] 2. $x^2-4x-2=6x-14-x^2$ [ANS] 3. $\\frac{1}{x+1}=\\frac{1}{2x-1}$ [ANS] 4. $x^2+3x+4=17x-5+x^2$\nHint: Study the", "answer_v1": [ "T", "F", "T", "T" ], "answer_type_v1": [ "TF", "TF", "TF", "TF" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Indicate with T (true) if the equations below are equivalent to a linear equation and with an F (False) if not. For example, the equation x^2+3x+4=x^2+5 turns into the linear equation 3x+4=5 after subtracting $x^2$ on both sides. Actually, the issue of whether a given equation is equivalent to a linear equation is quite subtle, but for the problems given here the answer will be (hopefully) obvious. To help you along, in this problem WeBWorK will indicate for each answer separately whether it's right or wrong and so you can get credit by just trying T's and F's. Of course you should think about the questions carefully and figure out why the answer is T or F. This will enable you to approach other questions more effectively. [ANS] 1. $x^2+3x+4=17x-5+x^2$ [ANS] 2. $\\frac{x}{6}-\\frac{2}{3}=\\frac{5}{6}$ [ANS] 3. $x^2-4x-2=6x-14-x^2$ [ANS] 4. $\\frac{1}{x+1}=\\frac{1}{2x-1}$\nHint: Study the", "answer_v2": [ "T", "T", "F", "T" ], "answer_type_v2": [ "TF", "TF", "TF", "TF" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Indicate with T (true) if the equations below are equivalent to a linear equation and with an F (False) if not. For example, the equation x^2+3x+4=x^2+5 turns into the linear equation 3x+4=5 after subtracting $x^2$ on both sides. Actually, the issue of whether a given equation is equivalent to a linear equation is quite subtle, but for the problems given here the answer will be (hopefully) obvious. To help you along, in this problem WeBWorK will indicate for each answer separately whether it's right or wrong and so you can get credit by just trying T's and F's. Of course you should think about the questions carefully and figure out why the answer is T or F. This will enable you to approach other questions more effectively. [ANS] 1. $x^2-4x-2=6x-14-x^2$ [ANS] 2. $\\frac{1}{x+1}=\\frac{1}{2x-1}$ [ANS] 3. $x^2+3x+4=17x-5+x^2$ [ANS] 4. $\\frac{x}{6}-\\frac{2}{3}=\\frac{5}{6}$\nHint: Study the", "answer_v3": [ "F", "T", "T", "T" ], "answer_type_v3": [ "TF", "TF", "TF", "TF" ], "options_v3": [ [], [], [], [] ] }, { "id": "Algebra_0229", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Linear equations", "level": "3", "keywords": [ "linear", "equations" ], "problem_v1": "Solve the equation for $A$:\nS=8 AB+6 BC-7 AC\n$\\begin{array}{cccc}\\hline & A=& & [ANS] [ANS] \\\\ \\hline \\end{array}$\nNote: Write your answer as a single fraction, and type the numerator and denominator separately in the answer blanks provided.", "answer_v1": [ "S-6*B*C", "8*B-7*C" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Solve the equation for $A$:\nS=2 AB+3 BC-9 AC\n$\\begin{array}{cccc}\\hline & A=& & [ANS] [ANS] \\\\ \\hline \\end{array}$\nNote: Write your answer as a single fraction, and type the numerator and denominator separately in the answer blanks provided.", "answer_v2": [ "S-3*B*C", "2*B-9*C" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Solve the equation for $A$:\nS=4 AB+4 BC+3 AC\n$\\begin{array}{cccc}\\hline & A=& & [ANS] [ANS] \\\\ \\hline \\end{array}$\nNote: Write your answer as a single fraction, and type the numerator and denominator separately in the answer blanks provided.", "answer_v3": [ "S-4*B*C", "4*B+3*C" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0230", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Linear equations", "level": "2", "keywords": [ "linear", "equations" ], "problem_v1": "Solve the equation for $y$:\n8x+6 y F=L\n$\\begin{array}{cccc}\\hline & y=& & [ANS] [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "L-8*x", "6*F" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Solve the equation for $y$:\n2x+3 y F=L\n$\\begin{array}{cccc}\\hline & y=& & [ANS] [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "L-2*x", "3*F" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Solve the equation for $y$:\n4x+4 y F=L\n$\\begin{array}{cccc}\\hline & y=& & [ANS] [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "L-4*x", "4*F" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0231", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Linear equations", "level": "2", "keywords": [ "linear", "equations" ], "problem_v1": "Solve the equation for $v$:\n k=\\frac{u}{3} \\left(v-9 J\\right)\n$\\begin{array}{cccc}\\hline & v=& & [ANS] [ANS] \\\\ \\hline \\end{array}$\nNote: Write your answer as a single fraction, and type the numerator and denominator separately in the answer blanks provided.", "answer_v1": [ "3*k+9*J*u", "u" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Solve the equation for $p$:\n s=\\frac{y}{4} \\left(p+7 C\\right)\n$\\begin{array}{cccc}\\hline & p=& & [ANS] [ANS] \\\\ \\hline \\end{array}$\nNote: Write your answer as a single fraction, and type the numerator and denominator separately in the answer blanks provided.", "answer_v2": [ "4*s-7*C*y", "y" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Solve the equation for $k$:\n u=\\frac{z}{3} \\left(k-9 M\\right)\n$\\begin{array}{cccc}\\hline & k=& & [ANS] [ANS] \\\\ \\hline \\end{array}$\nNote: Write your answer as a single fraction, and type the numerator and denominator separately in the answer blanks provided.", "answer_v3": [ "3*u+9*M*z", "z" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0232", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Linear inequalities", "level": "4", "keywords": [ "inequality", "linear", "interval notation", "subtract", "divide" ], "problem_v1": "You are riding in a taxi and can only pay with cash. You have to pay a flat fee of ${\\$35}$, and then pay ${\\$3.10}$ per mile. You have a total of ${\\$221}$ in your pocket. Let $x$ be the number of miles the taxi will drive you. You want to know how many miles you can afford. Write an inequality to represent this situation in terms of how many miles you can afford: Your inequality has three parts, a left side, a right side, and a comparison operator in the middle: $<$, $>$, $\\leq$, $\\geq$, $=$, or $\\neq$. Enter these as <, >, <=, >=,=, and!=. [ANS] [ANS] [ANS]\nSolve this inequality. At most how many miles can you afford?\nYou can afford at most [ANS] miles. Use interval notation to express the number of miles you can afford. [ANS]", "answer_v1": [ "(-infinity,60]", "60", "[0,60]" ], "answer_type_v1": [ "INT", "NV", "INT" ], "options_v1": [ [], [], [] ], "problem_v2": "You are riding in a taxi and can only pay with cash. You have to pay a flat fee of ${\\$20}$, and then pay ${\\$3.50}$ per mile. You have a total of ${\\$125}$ in your pocket. Let $x$ be the number of miles the taxi will drive you. You want to know how many miles you can afford. Write an inequality to represent this situation in terms of how many miles you can afford: Your inequality has three parts, a left side, a right side, and a comparison operator in the middle: $<$, $>$, $\\leq$, $\\geq$, $=$, or $\\neq$. Enter these as <, >, <=, >=,=, and!=. [ANS] [ANS] [ANS]\nSolve this inequality. At most how many miles can you afford?\nYou can afford at most [ANS] miles. Use interval notation to express the number of miles you can afford. [ANS]", "answer_v2": [ "(-infinity,30]", "30", "[0,30]" ], "answer_type_v2": [ "INT", "NV", "INT" ], "options_v2": [ [], [], [] ], "problem_v3": "You are riding in a taxi and can only pay with cash. You have to pay a flat fee of ${\\$25}$, and then pay ${\\$3.10}$ per mile. You have a total of ${\\$149}$ in your pocket. Let $x$ be the number of miles the taxi will drive you. You want to know how many miles you can afford. Write an inequality to represent this situation in terms of how many miles you can afford: Your inequality has three parts, a left side, a right side, and a comparison operator in the middle: $<$, $>$, $\\leq$, $\\geq$, $=$, or $\\neq$. Enter these as <, >, <=, >=,=, and!=. [ANS] [ANS] [ANS]\nSolve this inequality. At most how many miles can you afford?\nYou can afford at most [ANS] miles. Use interval notation to express the number of miles you can afford. [ANS]", "answer_v3": [ "(-infinity,40]", "40", "[0,40]" ], "answer_type_v3": [ "INT", "NV", "INT" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0234", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Linear inequalities", "level": "3", "keywords": [ "functions" ], "problem_v1": "The table below shows values of $v$ and the expressions $v-3$ and $2v-7$.\n$\\begin{array}{cccccccc}\\hline v=& 0 & 1 & 2 & 3 & 4 & 5 & 6 \\\\ \\hline v-3=&-3 &-2 &-1 & 0 & 1 & 2 & 3 \\\\ \\hline 2v-7=&-7 &-5 &-3 &-1 & 1 & 3 & 5 \\\\ \\hline \\end{array}$\nFor which values of $v$ in the table is:\n(a) $v-3 < 2v-7$? $v$=[ANS]\n(b) $v-3 > 2v-7$? $v$=[ANS]\n(c) $v-3=2v-7$? $v$=[ANS]\nIf there is more than one answer, enter your answers as a comma separated list. If there are no solutions, enter NONE.", "answer_v1": [ "(5, 6)", "(0, 1, 2, 3)", "4" ], "answer_type_v1": [ "UOL", "UOL", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The table below shows values of $v$ and the expressions $7-3v$ and $5-v$.\n$\\begin{array}{cccccccc}\\hline v=& 0 & 1 & 2 & 3 & 4 & 5 & 6 \\\\ \\hline 7-3v=& 7 & 4 & 1 &-2 &-5 &-8 &-11 \\\\ \\hline 5-v=& 5 & 4 & 3 & 2 & 1 & 0 &-1 \\\\ \\hline \\end{array}$\nFor which values of $v$ in the table is:\n(a) $7-3v < 5-v$? $v$=[ANS]\n(b) $7-3v > 5-v$? $v$=[ANS]\n(c) $7-3v=5-v$? $v$=[ANS]\nIf there is more than one answer, enter your answers as a comma separated list. If there are no solutions, enter NONE.", "answer_v2": [ "(2, 3, 4, 5, 6)", "0", "1" ], "answer_type_v2": [ "UOL", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The table below shows values of $v$ and the expressions $5-2v$ and $7-3v$.\n$\\begin{array}{cccccccc}\\hline v=& 0 & 1 & 2 & 3 & 4 & 5 & 6 \\\\ \\hline 5-2v=& 5 & 3 & 1 &-1 &-3 &-5 &-7 \\\\ \\hline 7-3v=& 7 & 4 & 1 &-2 &-5 &-8 &-11 \\\\ \\hline \\end{array}$\nFor which values of $v$ in the table is:\n(a) $5-2v < 7-3v$? $v$=[ANS]\n(b) $5-2v > 7-3v$? $v$=[ANS]\n(c) $5-2v=7-3v$? $v$=[ANS]\nIf there is more than one answer, enter your answers as a comma separated list. If there are no solutions, enter NONE.", "answer_v3": [ "(0, 1)", "(3, 4, 5, 6)", "2" ], "answer_type_v3": [ "UOL", "UOL", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0235", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Linear inequalities", "level": "3", "keywords": [ "algebra", "inequality", "temperature conversion" ], "problem_v1": "The temperatures for a 24-hour period ranged between $-2^\\circ F$ and $29^\\circ F$, inclusive. What was the range in Celsius degrees? (Use $F=\\frac{9}{5}C+32.$) Answer: [ANS]", "answer_v1": [ "[5/9*(-32+-2),5/9*(-32+29)]" ], "answer_type_v1": [ "INT" ], "options_v1": [ [] ], "problem_v2": "The temperatures for a 24-hour period ranged between $-8^\\circ F$ and $34^\\circ F$, inclusive. What was the range in Celsius degrees? (Use $F=\\frac{9}{5}C+32.$) Answer: [ANS]", "answer_v2": [ "[5/9*(-32+-8),5/9*(-32+34)]" ], "answer_type_v2": [ "INT" ], "options_v2": [ [] ], "problem_v3": "The temperatures for a 24-hour period ranged between $-6^\\circ F$ and $29^\\circ F$, inclusive. What was the range in Celsius degrees? (Use $F=\\frac{9}{5}C+32.$) Answer: [ANS]", "answer_v3": [ "[5/9*(-32+-6),5/9*(-32+29)]" ], "answer_type_v3": [ "INT" ], "options_v3": [ [] ] }, { "id": "Algebra_0236", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Graphs of lines", "level": "3", "keywords": [ "calculus", "algebra", "horizontal lines", "linear equations", "slope", "vertical lines" ], "problem_v1": "Determine whether there exists a constant $c$ such that the line $x+cy=5$ Has slope $5$: [ANS]\nPasses through $(2,3)$: [ANS]\nIs horizontal: [ANS]\nIs vertical: [ANS]\nNote: In each case, your answer is either the value of $c$ satisfying the requirement, or DNE when such a constant $c$ does not exist.", "answer_v1": [ "-(1/5)", "1", "DNE", "0" ], "answer_type_v1": [ "NV", "NV", "OE", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Determine whether there exists a constant $c$ such that the line $x+cy=-9$ Has slope $-3$: [ANS]\nPasses through $(9,-7)$: [ANS]\nIs horizontal: [ANS]\nIs vertical: [ANS]\nNote: In each case, your answer is either the value of $c$ satisfying the requirement, or DNE when such a constant $c$ does not exist.", "answer_v2": [ "1/3", "18/7", "DNE", "0" ], "answer_type_v2": [ "NV", "NV", "OE", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Determine whether there exists a constant $c$ such that the line $x+cy=-4$ Has slope $1$: [ANS]\nPasses through $(2,-5)$: [ANS]\nIs horizontal: [ANS]\nIs vertical: [ANS]\nNote: In each case, your answer is either the value of $c$ satisfying the requirement, or DNE when such a constant $c$ does not exist.", "answer_v3": [ "-1", "6/5", "DNE", "0" ], "answer_type_v3": [ "NV", "NV", "OE", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Algebra_0237", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "algebra", "inequalities" ], "problem_v1": "A car rental company offers the following two plans for renting a car: Plan A: 30 dollars per day and 18 cents per mile Plan B: 50 dollars per day with free unlimited mileage Q: How many miles must one drive in order to justify choosing Plan B? A: One must drive more than [ANS] miles to justify choosing Plan B.", "answer_v1": [ "111.111" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A car rental company offers the following two plans for renting a car: Plan A: 30 dollars per day and 11 cents per mile Plan B: 50 dollars per day with free unlimited mileage Q: How many miles must one drive in order to justify choosing Plan B? A: One must drive more than [ANS] miles to justify choosing Plan B.", "answer_v2": [ "181.818" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A car rental company offers the following two plans for renting a car: Plan A: 30 dollars per day and 14 cents per mile Plan B: 50 dollars per day with free unlimited mileage Q: How many miles must one drive in order to justify choosing Plan B? A: One must drive more than [ANS] miles to justify choosing Plan B.", "answer_v3": [ "142.857" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0238", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "solve", "inequality", "linear", "set notation", "interval notation", "compound", "fraction" ], "problem_v1": "A city\u2019s homeless population since the year 2000 can be modeled by this function:\n${f(x)={45\\!\\left(x-2000\\right)+540}}$ where $x$ represents the year, and $f(x)$ represents the number of homeless people in that year. Estimate the years when this city\u2019s homeless population will be between ${2565}$ and ${4140}$ people. Use an inequality to solve this problem. Solution: This city\u2019s homeless population will be between ${2565}$ and ${4140}$ people between the year [ANS] (smaller value) and the year [ANS] (bigger value).", "answer_v1": [ "2045", "2080" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A city\u2019s homeless population since the year 2000 can be modeled by this function:\n${f(x)={30\\!\\left(x-2000\\right)+590}}$ where $x$ represents the year, and $f(x)$ represents the number of homeless people in that year. Estimate the years when this city\u2019s homeless population will be between ${1490}$ and ${2540}$ people. Use an inequality to solve this problem. Solution: This city\u2019s homeless population will be between ${1490}$ and ${2540}$ people between the year [ANS] (smaller value) and the year [ANS] (bigger value).", "answer_v2": [ "2030", "2065" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A city\u2019s homeless population since the year 2000 can be modeled by this function:\n${f(x)={35\\!\\left(x-2000\\right)+540}}$ where $x$ represents the year, and $f(x)$ represents the number of homeless people in that year. Estimate the years when this city\u2019s homeless population will be between ${1765}$ and ${2990}$ people. Use an inequality to solve this problem. Solution: This city\u2019s homeless population will be between ${1765}$ and ${2990}$ people between the year [ANS] (smaller value) and the year [ANS] (bigger value).", "answer_v3": [ "2035", "2070" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0239", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "solve", "inequality", "linear", "set notation", "interval notation", "compound", "fraction" ], "problem_v1": "A rectangle\u2019s length is $4$ yards less than $5$ times of its width. If the rectangle\u2019s perimeter must be between $40$ and $100$ yards, what is the range of its width? Use an inequality to solve this problem. Solution: The rectangle\u2019s width must be between [ANS] yards (smaller value) and [ANS] yards (bigger value).", "answer_v1": [ "4", "9" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A rectangle\u2019s length is $6$ yards less than $2$ times of its width. If the rectangle\u2019s perimeter must be between $0$ and $30$ yards, what is the range of its width? Use an inequality to solve this problem. Solution: The rectangle\u2019s width must be between [ANS] yards (smaller value) and [ANS] yards (bigger value).", "answer_v2": [ "2", "7" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A rectangle\u2019s length is $5$ yards less than $3$ times of its width. If the rectangle\u2019s perimeter must be between $14$ and $54$ yards, what is the range of its width? Use an inequality to solve this problem. Solution: The rectangle\u2019s width must be between [ANS] yards (smaller value) and [ANS] yards (bigger value).", "answer_v3": [ "3", "8" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0240", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "calculator", "application" ], "problem_v1": "A new mine has $96000$ tons of coals in reserve. Every year, it plans to sell $4800$ tons. Define a function to model the amount of coals, and locate the points to answer the following questions:\nAfter $6$ years, there will be [ANS] tons of coals left.\nAfter [ANS] years, there will be ${62400}$ tons of coals left.\nAt this rate, coals in the mine will be used up in [ANS] years.", "answer_v1": [ "67200", "7", "20" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A new mine has $57000$ tons of coals in reserve. Every year, it plans to sell $3000$ tons. Define a function to model the amount of coals, and locate the points to answer the following questions:\nAfter $9$ years, there will be [ANS] tons of coals left.\nAfter [ANS] years, there will be ${39000}$ tons of coals left.\nAt this rate, coals in the mine will be used up in [ANS] years.", "answer_v2": [ "30000", "6", "19" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A new mine has $60000$ tons of coals in reserve. Every year, it plans to sell $3000$ tons. Define a function to model the amount of coals, and locate the points to answer the following questions:\nAfter $9$ years, there will be [ANS] tons of coals left.\nAfter [ANS] years, there will be ${42000}$ tons of coals left.\nAt this rate, coals in the mine will be used up in [ANS] years.", "answer_v3": [ "33000", "6", "20" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0241", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "linear", "equation", "slope", "slope-intercept", "application" ], "problem_v1": "A company set aside a certain amount of money in the year 2000. The company spent exactly the same amount from that fund each year on perks for its employees. In $2003$, there was still ${\\$704{,}000}$ left in the fund. In $2007$, there was ${\\$532{,}000}$ left. Let $x$ be the number of years since 2000, and let $y$ be the amount of money left in the fund that year. Use a linear equation to model the amount of money left in the fund after so many years.\nThe linear model\u2019s slope-intercept equation is [ANS].\nIn the year $2009$, there was $[ANS] left in the fund.\nIn the year [ANS], the fund will be empty.", "answer_v1": [ "y = -43000*x+833000", "446000", "2020" ], "answer_type_v1": [ "EQ", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A company set aside a certain amount of money in the year 2000. The company spent exactly the same amount from that fund each year on perks for its employees. In $2002$, there was still ${\\$929{,}000}$ left in the fund. In $2006$, there was ${\\$841{,}000}$ left. Let $x$ be the number of years since 2000, and let $y$ be the amount of money left in the fund that year. Use a linear equation to model the amount of money left in the fund after so many years.\nThe linear model\u2019s slope-intercept equation is [ANS].\nIn the year $2011$, there was $[ANS] left in the fund.\nIn the year [ANS], the fund will be empty.", "answer_v2": [ "y = -22000*x+973000", "731000", "2045" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A company set aside a certain amount of money in the year 2000. The company spent exactly the same amount from that fund each year on perks for its employees. In $2002$, there was still ${\\$784{,}000}$ left in the fund. In $2006$, there was ${\\$668{,}000}$ left. Let $x$ be the number of years since 2000, and let $y$ be the amount of money left in the fund that year. Use a linear equation to model the amount of money left in the fund after so many years.\nThe linear model\u2019s slope-intercept equation is [ANS].\nIn the year $2009$, there was $[ANS] left in the fund.\nIn the year [ANS], the fund will be empty.", "answer_v3": [ "y = -29000*x+842000", "581000", "2030" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0242", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "rate", "ratio", "proportion", "solve", "word problem" ], "problem_v1": "Fabrienne can type $260$ words in $3.5$ minutes. At this rate, how many words can she type in $24.5$ minutes?\nFabrienne can type [ANS] words in $24.5$ minutes.", "answer_v1": [ "1820" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Tien can type $200$ words in $4.5$ minutes. At this rate, how many words can he type in $18$ minutes?\nTien can type [ANS] words in $18$ minutes.", "answer_v2": [ "800" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Dawn can type $220$ words in $3.5$ minutes. At this rate, how many words can she type in $17.5$ minutes?\nDawn can type [ANS] words in $17.5$ minutes.", "answer_v3": [ "1100" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0243", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "solve", "linear", "equation", "application", "add", "divide" ], "problem_v1": "A school purchased a batch of T-shirts from a company. The company charged ${\\$9}$ per T-shirt, and gave the school a ${\\$80}$ rebate. If the school had a net expense of ${\\$3{,}790}$ from the purchase, how many T-shirts did the school buy?\nThe school purchased [ANS] T-shirts.", "answer_v1": [ "430" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A school purchased a batch of T-shirts from a company. The company charged ${\\$3}$ per T-shirt, and gave the school a ${\\$100}$ rebate. If the school had a net expense of ${\\$890}$ from the purchase, how many T-shirts did the school buy?\nThe school purchased [ANS] T-shirts.", "answer_v2": [ "330" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A school purchased a batch of T-shirts from a company. The company charged ${\\$5}$ per T-shirt, and gave the school a ${\\$80}$ rebate. If the school had a net expense of ${\\$1{,}670}$ from the purchase, how many T-shirts did the school buy?\nThe school purchased [ANS] T-shirts.", "answer_v3": [ "350" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0244", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "solve", "linear", "equation", "application", "subtract", "divide" ], "problem_v1": "Eric has ${\\$85}$ in his piggy bank. He plans to purchase some Pokemon cards, which costs ${\\$2.65}$ each. He plans to save ${\\$42.60}$ to purchase another toy. At most how many Pokemon cards can he purchase?\nEric can purchase at most [ANS] Pokemon cards.", "answer_v1": [ "16" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Hannah has ${\\$71}$ in her piggy bank. She plans to purchase some Pokemon cards, which costs ${\\$2.15}$ each. She plans to save ${\\$49.50}$ to purchase another toy. At most how many Pokemon cards can he purchase?\nHannah can purchase at most [ANS] Pokemon cards.", "answer_v2": [ "10" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evan has ${\\$76}$ in his piggy bank. He plans to purchase some Pokemon cards, which costs ${\\$2.25}$ each. He plans to save ${\\$46.75}$ to purchase another toy. At most how many Pokemon cards can he purchase?\nEvan can purchase at most [ANS] Pokemon cards.", "answer_v3": [ "13" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0245", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "linear", "equation", "slope", "slope-intercept", "application" ], "problem_v1": "Scientists are conducting an experiment with a gas in a sealed container. The mass of the gas is measured, and the scientists realize that the gas is leaking over time in a linear way. Its mass is leaking by $6.2$ grams per minute. Eight minutes since the experiment started, the remaining gas had a mass of $248$ grams. Let $x$ be the number of minutes that have passed since the experiment started, and let $y$ be the mass of the gas in grams at that moment. Use a linear equation to model the weight of the gas over time.\nThis line\u2019s slope-intercept equation is [ANS].\n$33$ minutes after the experiment started, there would be [ANS] grams of gas left.\nIf a linear model continues to be accurate, [ANS] minutes since the experiment started, all gas in the container will be gone.", "answer_v1": [ "y = -6.2*x+297.6", "93", "48" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Scientists are conducting an experiment with a gas in a sealed container. The mass of the gas is measured, and the scientists realize that the gas is leaking over time in a linear way. Its mass is leaking by $9.3$ grams per minute. Five minutes since the experiment started, the remaining gas had a mass of $325.5$ grams. Let $x$ be the number of minutes that have passed since the experiment started, and let $y$ be the mass of the gas in grams at that moment. Use a linear equation to model the weight of the gas over time.\nThis line\u2019s slope-intercept equation is [ANS].\n$40$ minutes after the experiment started, there would be [ANS] grams of gas left.\nIf a linear model continues to be accurate, [ANS] minutes since the experiment started, all gas in the container will be gone.", "answer_v2": [ "y = -9.3*x+372", "0", "40" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Scientists are conducting an experiment with a gas in a sealed container. The mass of the gas is measured, and the scientists realize that the gas is leaking over time in a linear way. Its mass is leaking by $6.4$ grams per minute. Six minutes since the experiment started, the remaining gas had a mass of $236.8$ grams. Let $x$ be the number of minutes that have passed since the experiment started, and let $y$ be the mass of the gas in grams at that moment. Use a linear equation to model the weight of the gas over time.\nThis line\u2019s slope-intercept equation is [ANS].\n$32$ minutes after the experiment started, there would be [ANS] grams of gas left.\nIf a linear model continues to be accurate, [ANS] minutes since the experiment started, all gas in the container will be gone.", "answer_v3": [ "y = -6.4*x+275.2", "70.4", "43" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0246", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "rate", "ratio", "proportion", "solve", "word problem" ], "problem_v1": "A car insurance company charges insurance premium in proportion to the car\u2019s value. For a ${\\$19{,}000}$ car, the monthly premium is ${\\$75}$. How much premium would you expect to pay if your car is worth ${\\$47{,}000}$?\nA ${\\$47{,}000}$ car would be charged a monthly premium of $ [ANS].", "answer_v1": [ "185.53" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A car insurance company charges insurance premium in proportion to the car\u2019s value. For a ${\\$11{,}000}$ car, the monthly premium is ${\\$90}$. How much premium would you expect to pay if your car is worth ${\\$42{,}000}$?\nA ${\\$42{,}000}$ car would be charged a monthly premium of $ [ANS].", "answer_v2": [ "343.64" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A car insurance company charges insurance premium in proportion to the car\u2019s value. For a ${\\$14{,}000}$ car, the monthly premium is ${\\$75}$. How much premium would you expect to pay if your car is worth ${\\$44{,}000}$?\nA ${\\$44{,}000}$ car would be charged a monthly premium of $ [ANS].", "answer_v3": [ "235.71" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0247", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "solve", "linear", "equation", "application", "subtract", "divide" ], "problem_v1": "Fabrienne hired a face-painter for a birthday party. The painter charged a flat fee of ${\\$90}$, and then charged ${\\$4.50}$ per person. In the end, Fabrienne paid a total of ${\\$193.50}$. How many people used the face-painter\u2019s service? [ANS] people used the face-painter\u2019s service.", "answer_v1": [ "23" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Tien hired a face-painter for a birthday party. The painter charged a flat fee of ${\\$50}$, and then charged ${\\$5.50}$ per person. In the end, Tien paid a total of ${\\$121.50}$. How many people used the face-painter\u2019s service? [ANS] people used the face-painter\u2019s service.", "answer_v2": [ "13" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Dawn hired a face-painter for a birthday party. The painter charged a flat fee of ${\\$65}$, and then charged ${\\$4.50}$ per person. In the end, Dawn paid a total of ${\\$132.50}$. How many people used the face-painter\u2019s service? [ANS] people used the face-painter\u2019s service.", "answer_v3": [ "15" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0248", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "rate", "ratio", "proportion", "solve", "word problem" ], "problem_v1": "Roses are on sale! A dozen roses cost ${\\$11.99}$. Cody wants to buy $19$ roses. What\u2019s a fair price to charge?\nCody should pay $ [ANS] for $19$ roses.", "answer_v1": [ "18.98" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Roses are on sale! A dozen roses cost ${\\$7.99}$. Elishua wants to buy $23$ roses. What\u2019s a fair price to charge?\nElishua should pay $ [ANS] for $23$ roses.", "answer_v2": [ "15.31" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Roses are on sale! A dozen roses cost ${\\$8.99}$. Kylie wants to buy $19$ roses. What\u2019s a fair price to charge?\nKylie should pay $ [ANS] for $19$ roses.", "answer_v3": [ "14.23" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0249", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "solve", "linear", "equation", "application", "subtract", "divide" ], "problem_v1": "A gym charges members ${\\$35}$ for a registration fee, and then ${\\$32}$ per month. You became a member some time ago, and now you have paid a total of ${\\$547}$ to the gym. How many months have passed since you joined the gym? [ANS] months have passed since you joined the gym.", "answer_v1": [ "16" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A gym charges members ${\\$20}$ for a registration fee, and then ${\\$40}$ per month. You became a member some time ago, and now you have paid a total of ${\\$380}$ to the gym. How many months have passed since you joined the gym? [ANS] months have passed since you joined the gym.", "answer_v2": [ "9" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A gym charges members ${\\$25}$ for a registration fee, and then ${\\$32}$ per month. You became a member some time ago, and now you have paid a total of ${\\$377}$ to the gym. How many months have passed since you joined the gym? [ANS] months have passed since you joined the gym.", "answer_v3": [ "11" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0250", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "linear", "equation", "slope", "slope-intercept", "application" ], "problem_v1": "A biologist has been observing a tree\u2019s height. $13$ months into the observation, the tree was $18.95$ feet tall. $19$ months into the observation, the tree was $20.45$ feet tall. Let $x$ be the number of months passed since the observations started, and let $y$ be the tree\u2019s height at that time. Use a linear equation to model the tree\u2019s height as the number of months pass.\nThis line\u2019s slope-intercept equation is [ANS].\n$26$ months after the observations started, the tree would be [ANS] feet in height. [ANS] months after the observation started, the tree would be $28.95$ feet tall.", "answer_v1": [ "y = 0.25*x+15.7", "22.2", "53" ], "answer_type_v1": [ "EQ", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A biologist has been observing a tree\u2019s height. $10$ months into the observation, the tree was $19.6$ feet tall. $17$ months into the observation, the tree was $20.37$ feet tall. Let $x$ be the number of months passed since the observations started, and let $y$ be the tree\u2019s height at that time. Use a linear equation to model the tree\u2019s height as the number of months pass.\nThis line\u2019s slope-intercept equation is [ANS].\n$30$ months after the observations started, the tree would be [ANS] feet in height. [ANS] months after the observation started, the tree would be $24.33$ feet tall.", "answer_v2": [ "y = 0.11*x+18.5", "21.8", "53" ], "answer_type_v2": [ "EQ", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A biologist has been observing a tree\u2019s height. $11$ months into the observation, the tree was $17.66$ feet tall. $18$ months into the observation, the tree was $18.78$ feet tall. Let $x$ be the number of months passed since the observations started, and let $y$ be the tree\u2019s height at that time. Use a linear equation to model the tree\u2019s height as the number of months pass.\nThis line\u2019s slope-intercept equation is [ANS].\n$26$ months after the observations started, the tree would be [ANS] feet in height. [ANS] months after the observation started, the tree would be $24.38$ feet tall.", "answer_v3": [ "y = 0.16*x+15.9", "20.06", "53" ], "answer_type_v3": [ "EQ", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0251", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "linear functions", "correlation coefficient", "least squares", "regression" ], "problem_v1": "The table below shows the IQ of ten students and the number of hours of TV each watches per week.\n$\\begin{array}{ccccccccccc}\\hline IQ & 110 & 105 & 120 & 140 & 100 & 125 & 130 & 105 & 115 & 110 \\\\ \\hline TV & 11 & 12 & 9 & 3 & 11 & 9 & 5 & 6 & 12 & 3 \\\\ \\hline \\end{array}$\n(a) Use a calculator or computer to find the least squares regression line, $y=mx+b$, where the dependent variable $y$ denotes the number of hours of TV watched as a function of IQ, $x$. Enter the equation for the line below. $y=$ [ANS]\n(b) What is the correlation coefficient of the regression line? $r=$ [ANS]\n(Enter $r$ accurate to at least four decimal places.)", "answer_v1": [ "25.0972+-0.1465*x", "-0.5187" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The table below shows the IQ of ten students and the number of hours of TV each watches per week.\n$\\begin{array}{ccccccccccc}\\hline IQ & 110 & 105 & 120 & 140 & 100 & 125 & 130 & 105 & 115 & 110 \\\\ \\hline TV & 8 & 14 & 6 & 3 & 14 & 9 & 3 & 5 & 13 & 1 \\\\ \\hline \\end{array}$\n(a) Use a calculator or computer to find the least squares regression line, $y=mx+b$, where the dependent variable $y$ denotes the number of hours of TV watched as a function of IQ, $x$. Enter the equation for the line below. $y=$ [ANS]\n(b) What is the correlation coefficient of the regression line? $r=$ [ANS]\n(Enter $r$ accurate to at least four decimal places.)", "answer_v2": [ "30.2361+-0.1951*x", "-0.513" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The table below shows the IQ of ten students and the number of hours of TV each watches per week.\n$\\begin{array}{ccccccccccc}\\hline IQ & 110 & 105 & 120 & 140 & 100 & 125 & 130 & 105 & 115 & 110 \\\\ \\hline TV & 9 & 13 & 7 & 2 & 11 & 9 & 7 & 8 & 15 & 2 \\\\ \\hline \\end{array}$\n(a) Use a calculator or computer to find the least squares regression line, $y=mx+b$, where the dependent variable $y$ denotes the number of hours of TV watched as a function of IQ, $x$. Enter the equation for the line below. $y=$ [ANS]\n(b) What is the correlation coefficient of the regression line? $r=$ [ANS]\n(Enter $r$ accurate to at least four decimal places.)", "answer_v3": [ "27.0694+-0.1618*x", "-0.4883" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0252", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "linear functions", "correlation coefficient", "least squares", "regression" ], "problem_v1": "The table below gives the data on hand strength collected from college freshman using a grip meter. The preferred row gives the hand strength of each student's preferred hand while the nonpreferred column gives the hand strength of their nonpreferred hand.\n$\\begin{array}{ccccccccccc}\\hline Preferred & 53 & 26 & 52 & 47 & 21 & 30 & 32 & 20 & 28 & 40 \\\\ \\hline Nonpreferred & 48 & 20 & 45 & 43 & 21 & 26 & 28 & 21 & 25 & 36 \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccccc}\\hline Preferred & 49 & 45 & 54 & 25 & 39 & 52 & 27 & 45 & 32 & 28 \\\\ \\hline Nonpreferred & 41 & 43 & 50 & 26 & 32 & 45 & 26 & 43 & 26 & 26 \\\\ \\hline \\end{array}$\n(a) Use a calculator or computer to find the least squares regression line, $y=mx+b$, where the independent variable $x$ denotes the grip strength of the preferred hand and the dependent variable $y$ denotes the grip strength of the nonpreferred hand. Enter the equation for the line below. (Use at least four decimal places in any constants in your answer.) $y=$ [ANS]\n(b) What would the predicted grip strength in the nonpreferred hand be for a student with a preferred hand strength of 35 kg? strength=[ANS] kg (round to nearest whole number) (c) For each of the preferred grip strengths values, decided whether you would use extrapolation or interpolation to approximate the value of the corresponding nonpreferred grip strength. (i) A preferred grip strength of 15 kg? [ANS] (ii) A preferred grip strength of 35 kg? [ANS] (iii) A preferred grip strength of 57 kg? [ANS] (iv) A preferred grip strength of 28 kg? [ANS]", "answer_v1": [ "1.6105+0.8574*x", "32", "Extrapolation", "Interpolation", "Extrapolation", "Interpolation" ], "answer_type_v1": [ "EX", "NV", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [], [], [ "Interpolation", "Extrapolation" ], [ "Interpolation", "Extrapolation" ], [ "Interpolation", "Extrapolation" ], [ "Interpolation", "Extrapolation" ] ], "problem_v2": "The table below gives the data on hand strength collected from college freshman using a grip meter. The preferred row gives the hand strength of each student's preferred hand while the nonpreferred column gives the hand strength of their nonpreferred hand.\n$\\begin{array}{ccccccccccc}\\hline Preferred & 45 & 20 & 27 & 52 & 52 & 40 & 30 & 54 & 53 & 26 \\\\ \\hline Nonpreferred & 40 & 19 & 27 & 47 & 47 & 40 & 27 & 48 & 48 & 19 \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccccc}\\hline Preferred & 32 & 49 & 45 & 32 & 28 & 21 & 28 & 25 & 39 & 47 \\\\ \\hline Nonpreferred & 29 & 44 & 41 & 26 & 21 & 19 & 24 & 26 & 30 & 43 \\\\ \\hline \\end{array}$\n(a) Use a calculator or computer to find the least squares regression line, $y=mx+b$, where the independent variable $x$ denotes the grip strength of the preferred hand and the dependent variable $y$ denotes the grip strength of the nonpreferred hand. Enter the equation for the line below. (Use at least four decimal places in any constants in your answer.) $y=$ [ANS]\n(b) What would the predicted grip strength in the nonpreferred hand be for a student with a preferred hand strength of 34 kg? strength=[ANS] kg (round to nearest whole number) (c) For each of the preferred grip strengths values, decided whether you would use extrapolation or interpolation to approximate the value of the corresponding nonpreferred grip strength. (i) A preferred grip strength of 16 kg? [ANS] (ii) A preferred grip strength of 28 kg? [ANS] (iii) A preferred grip strength of 34 kg? [ANS] (iv) A preferred grip strength of 56 kg? [ANS]", "answer_v2": [ "-0.7421+0.9125*x", "30", "Extrapolation", "Interpolation", "Interpolation", "Extrapolation" ], "answer_type_v2": [ "EX", "NV", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [], [], [ "Interpolation", "Extrapolation" ], [ "Interpolation", "Extrapolation" ], [ "Interpolation", "Extrapolation" ], [ "Interpolation", "Extrapolation" ] ], "problem_v3": "The table below gives the data on hand strength collected from college freshman using a grip meter. The preferred row gives the hand strength of each student's preferred hand while the nonpreferred column gives the hand strength of their nonpreferred hand.\n$\\begin{array}{ccccccccccc}\\hline Preferred & 39 & 21 & 54 & 30 & 26 & 25 & 52 & 53 & 32 & 49 \\\\ \\hline Nonpreferred & 37 & 20 & 52 & 27 & 21 & 26 & 49 & 46 & 29 & 37 \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccccc}\\hline Preferred & 27 & 47 & 32 & 52 & 45 & 28 & 20 & 28 & 45 & 40 \\\\ \\hline Nonpreferred & 26 & 43 & 26 & 45 & 41 & 27 & 20 & 22 & 43 & 36 \\\\ \\hline \\end{array}$\n(a) Use a calculator or computer to find the least squares regression line, $y=mx+b$, where the independent variable $x$ denotes the grip strength of the preferred hand and the dependent variable $y$ denotes the grip strength of the nonpreferred hand. Enter the equation for the line below. (Use at least four decimal places in any constants in your answer.) $y=$ [ANS]\n(b) What would the predicted grip strength in the nonpreferred hand be for a student with a preferred hand strength of 35 kg? strength=[ANS] kg (round to nearest whole number) (c) For each of the preferred grip strengths values, decided whether you would use extrapolation or interpolation to approximate the value of the corresponding nonpreferred grip strength. (i) A preferred grip strength of 16 kg? [ANS] (ii) A preferred grip strength of 56 kg? [ANS] (iii) A preferred grip strength of 27 kg? [ANS] (iv) A preferred grip strength of 35 kg? [ANS]", "answer_v3": [ "1.1918+0.8714*x", "32", "Extrapolation", "Extrapolation", "Interpolation", "Interpolation" ], "answer_type_v3": [ "EX", "NV", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [], [], [ "Interpolation", "Extrapolation" ], [ "Interpolation", "Extrapolation" ], [ "Interpolation", "Extrapolation" ], [ "Interpolation", "Extrapolation" ] ] }, { "id": "Algebra_0253", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "1", "keywords": [ "functions", "definition of function", "function notation" ], "problem_v1": "The sales tax on an item is 8\\%. Express the total cost, $C$, in terms of the price of the item, $P$. Be sure to use the correct case of the variable when entering your expression. $C=$ [ANS] $P$", "answer_v1": [ "1.08" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The sales tax on an item is 5\\%. Express the total cost, $C$, in terms of the price of the item, $P$. Be sure to use the correct case of the variable when entering your expression. $C=$ [ANS] $P$", "answer_v2": [ "1.05" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The sales tax on an item is 6\\%. Express the total cost, $C$, in terms of the price of the item, $P$. Be sure to use the correct case of the variable when entering your expression. $C=$ [ANS] $P$", "answer_v3": [ "1.06" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0254", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "functions", "average rate of change", "rate of change" ], "problem_v1": "In 2005, you have 55 CDs in your collection. In 2008, you have 140 CDs. In 2012, you have 45 CDs. What is the average rate of change in the size of your CD collection between:\n(a) 2005 and 2008? [ANS]\n(b) 2008 and 2012? [ANS]\n(c) 2005 and 2012? [ANS]", "answer_v1": [ "85/3", "-95/4", "-10/7" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "In 2005, you have 30 CDs in your collection. In 2008, you have 160 CDs. In 2012, you have 30 CDs. What is the average rate of change in the size of your CD collection between:\n(a) 2005 and 2008? [ANS]\n(b) 2008 and 2012? [ANS]\n(c) 2005 and 2012? [ANS]", "answer_v2": [ "130/3", "-130/4", "0/7" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "In 2005, you have 40 CDs in your collection. In 2008, you have 140 CDs. In 2012, you have 35 CDs. What is the average rate of change in the size of your CD collection between:\n(a) 2005 and 2008? [ANS]\n(b) 2008 and 2012? [ANS]\n(c) 2005 and 2012? [ANS]", "answer_v3": [ "100/3", "-105/4", "-5/7" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0255", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "lines", "linear functions", "slope-intercept", "point-slope form" ], "problem_v1": "A flight costs \\$11,500 to operate, regardless of the number of passengers. Each ticket costs \\$129. Express profit, $P$, as a linear function of the number of passengers, $n$, on the flight. $P=$ [ANS] (do not enter commas in your formula)", "answer_v1": [ "129*n-11500" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "A flight costs \\$8,500 to operate, regardless of the number of passengers. Each ticket costs \\$139. Express profit, $P$, as a linear function of the number of passengers, $n$, on the flight. $P=$ [ANS] (do not enter commas in your formula)", "answer_v2": [ "139*n-8500" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "A flight costs \\$9,500 to operate, regardless of the number of passengers. Each ticket costs \\$129. Express profit, $P$, as a linear function of the number of passengers, $n$, on the flight. $P=$ [ANS] (do not enter commas in your formula)", "answer_v3": [ "129*n-9500" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0257", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "rate of change", "lines", "linear functions", "slope", "intercept" ], "problem_v1": "The table below shows the cost $C$ to Company A, in dollars, of selling $x$ cups of coffee per day from a cart.\n$\\begin{array}{ccccccc}\\hline x & 0 & 5 & 10 & 50 & 100 & 200 \\\\ \\hline C & 60 & 61.25 & 62.5 & 72.5 & 85 & [ANS] \\\\ \\hline \\end{array}$\na) Assuming the function $C$ is linear. What is the slope of the line corresponding to the graph of $y=C(x)$? The line has slope=[ANS]\nb) The value for $C(200)$ is missing in the table above. In the blank provided in the table, enter the correct value for $C(200)$ assuming the cost is a linear function of $x$. Now consider Company B, whose cost $F$, in dollars, of selling $x$ cups of coffee per day from a cart is given in the table below:\n$\\begin{array}{ccccccc}\\hline x & 0 & 100 & 150 & 175 & 190 & 200 \\\\ \\hline F & 55 & 75 & 85 & 90 & 93 & 95 \\\\ \\hline \\end{array}$\nc) Which company pays higher fixed costs for rent and labor? [ANS] A. Company A B. Company B\nd) Which graph will be steeper, the graph corresponding to Company A's cost C(x) C(x), or the graph corresponding to Company B's cost F(x) F(x) [ANS] A. Company A B. Company B", "answer_v1": [ "110", "0.25", "A", "A" ], "answer_type_v1": [ "NV", "NV", "MCS", "MCS" ], "options_v1": [ [], [], [ "A", "B" ], [ "A", "B" ] ], "problem_v2": "The table below shows the cost $C$ to Company A, in dollars, of selling $x$ cups of coffee per day from a cart.\n$\\begin{array}{ccccccc}\\hline x & 0 & 5 & 10 & 50 & 100 & 200 \\\\ \\hline C & 30 & 31.25 & 32.5 & 42.5 & 55 & [ANS] \\\\ \\hline \\end{array}$\na) Assuming the function $C$ is linear. What is the slope of the line corresponding to the graph of $y=C(x)$? The line has slope=[ANS]\nb) The value for $C(200)$ is missing in the table above. In the blank provided in the table, enter the correct value for $C(200)$ assuming the cost is a linear function of $x$. Now consider Company B, whose cost $F$, in dollars, of selling $x$ cups of coffee per day from a cart is given in the table below:\n$\\begin{array}{ccccccc}\\hline x & 0 & 100 & 150 & 175 & 190 & 200 \\\\ \\hline F & 35 & 45 & 50 & 52.5 & 54 & 55 \\\\ \\hline \\end{array}$\nc) Which company pays higher fixed costs for rent and labor? [ANS] A. Company A B. Company B\nd) Which graph will be steeper, the graph corresponding to Company A's cost C(x) C(x), or the graph corresponding to Company B's cost F(x) F(x) [ANS] A. Company A B. Company B", "answer_v2": [ "80", "0.25", "B", "A" ], "answer_type_v2": [ "NV", "NV", "MCS", "MCS" ], "options_v2": [ [], [], [ "A", "B" ], [ "A", "B" ] ], "problem_v3": "The table below shows the cost $C$ to Company A, in dollars, of selling $x$ cups of coffee per day from a cart.\n$\\begin{array}{ccccccc}\\hline x & 0 & 5 & 10 & 50 & 100 & 200 \\\\ \\hline C & 40 & 41.25 & 42.5 & 52.5 & 65 & [ANS] \\\\ \\hline \\end{array}$\na) Assuming the function $C$ is linear. What is the slope of the line corresponding to the graph of $y=C(x)$? The line has slope=[ANS]\nb) The value for $C(200)$ is missing in the table above. In the blank provided in the table, enter the correct value for $C(200)$ assuming the cost is a linear function of $x$. Now consider Company B, whose cost $F$, in dollars, of selling $x$ cups of coffee per day from a cart is given in the table below:\n$\\begin{array}{ccccccc}\\hline x & 0 & 100 & 150 & 175 & 190 & 200 \\\\ \\hline F & 45 & 65 & 75 & 80 & 83 & 85 \\\\ \\hline \\end{array}$\nc) Which company pays higher fixed costs for rent and labor? [ANS] A. Company A B. Company B\nd) Which graph will be steeper, the graph corresponding to Company A's cost C(x) C(x), or the graph corresponding to Company B's cost F(x) F(x) [ANS] A. Company A B. Company B", "answer_v3": [ "90", "0.25", "B", "A" ], "answer_type_v3": [ "NV", "NV", "MCS", "MCS" ], "options_v3": [ [], [], [ "A", "B" ], [ "A", "B" ] ] }, { "id": "Algebra_0258", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "rate of change", "lines", "linear functions", "slope", "intercept" ], "problem_v1": "You bought a new car for \\$25,500 in 2005, and the value of the car depreciates by \\$700 each year. Find a formula for $V$, the value of the car, in terms of $t$, the number of years since 2005. $V(t)=\\ $ [ANS]\n(Be sure NOT TO USE ANY COMMAS when you enter your formula. For example enter two thousand as 2000 and not as 2,000.) (Be sure NOT TO USE ANY COMMAS when you enter your formula. For example enter two thousand as 2000 and not as 2,000.)", "answer_v1": [ "25500-700*t" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "You bought a new car for \\$16,500 in 2005, and the value of the car depreciates by \\$900 each year. Find a formula for $V$, the value of the car, in terms of $t$, the number of years since 2005. $V(t)=\\ $ [ANS]\n(Be sure NOT TO USE ANY COMMAS when you enter your formula. For example enter two thousand as 2000 and not as 2,000.) (Be sure NOT TO USE ANY COMMAS when you enter your formula. For example enter two thousand as 2000 and not as 2,000.)", "answer_v2": [ "16500-900*t" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "You bought a new car for \\$19,500 in 2005, and the value of the car depreciates by \\$700 each year. Find a formula for $V$, the value of the car, in terms of $t$, the number of years since 2005. $V(t)=\\ $ [ANS]\n(Be sure NOT TO USE ANY COMMAS when you enter your formula. For example enter two thousand as 2000 and not as 2,000.) (Be sure NOT TO USE ANY COMMAS when you enter your formula. For example enter two thousand as 2000 and not as 2,000.)", "answer_v3": [ "19500-700*t" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0259", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "rate of change", "lines", "linear functions", "slope", "intercept" ], "problem_v1": "Steadia is an island which experienced approximately linear population growth from 1950 to 2000. On the other hand, Randomian has experienced some turmoil more recently and did not experience linear nor near-linear growth.\n$\\begin{array}{ccccccc}\\hline Year & 1950 & 1960 & 1970 & 1980 & 1990 & 2000 \\\\ \\hline Pop. of country A & 7.7 & 10.1 & 12.7 & 15.1 & 17.3 & 20 \\\\ \\hline Pop. of country B & 8.3 & 10.5 & 12.3 & 13.9 & 13.4 & 20.1 \\\\ \\hline \\end{array}$\na) The table above gives the population of these two countries, in millions. Does country A or country B represent the population of Steadia? Enter just the letter of the country in the blank (A or B). [ANS]\nb) What is the approximate rate of change of the linear function? [ANS]\nc) What does the rate of change in your answer to (b) represent in practical terms? (select all that apply) [ANS] A. The amount (in millions) the population grows each year. B. The amount (in millions) the population grows every 10 years. C. The number of years it takes for the population to increase by one million people. D. The percent the population grows each year. E. The total amount (in millions) the population grows from 1950 to 2000. F. None of the above.\nd) Estimate the population of Steadia in 1986. [ANS] million people", "answer_v1": [ "A", "0.24", "A", "16.54" ], "answer_type_v1": [ "MCS", "NV", "MCS", "NV" ], "options_v1": [ [ "A", "B" ], [], [ "A", "B", "C", "D", "E", "F" ], [] ], "problem_v2": "Steadia is an island which experienced approximately linear population growth from 1950 to 2000. On the other hand, Randomian has experienced some turmoil more recently and did not experience linear nor near-linear growth.\n$\\begin{array}{ccccccc}\\hline Year & 1950 & 1960 & 1970 & 1980 & 1990 & 2000 \\\\ \\hline Pop. of country A & 6.9 & 8.4 & 9.6 & 11.2 & 12.7 & 14.5 \\\\ \\hline Pop. of country B & 8.9 & 10.5 & 12.3 & 14.5 & 14.1 & 20 \\\\ \\hline \\end{array}$\na) The table above gives the population of these two countries, in millions. Does country A or country B represent the population of Steadia? Enter just the letter of the country in the blank (A or B). [ANS]\nb) What is the approximate rate of change of the linear function? [ANS]\nc) What does the rate of change in your answer to (b) represent in practical terms? (select all that apply) [ANS] A. The number of years it takes for the population to increase by one million people. B. The amount (in millions) the population grows each year. C. The amount (in millions) the population grows every 10 years. D. The total amount (in millions) the population grows from 1950 to 2000. E. The percent the population grows each year. F. None of the above.\nd) Estimate the population of Steadia in 1986. [ANS] million people", "answer_v2": [ "A", "0.15", "B", "12.1" ], "answer_type_v2": [ "MCS", "NV", "MCS", "NV" ], "options_v2": [ [ "A", "B" ], [], [ "A", "B", "C", "D", "E", "F" ], [] ], "problem_v3": "Steadia is an island which experienced approximately linear population growth from 1950 to 2000. On the other hand, Randomian has experienced some turmoil more recently and did not experience linear nor near-linear growth.\n$\\begin{array}{ccccccc}\\hline Year & 1950 & 1960 & 1970 & 1980 & 1990 & 2000 \\\\ \\hline Pop. of country A & 9.7 & 12.1 & 14.3 & 15.9 & 15.2 & 22.7 \\\\ \\hline Pop. of country B & 8.7 & 10.5 & 12.5 & 14.3 & 15.9 & 17.8 \\\\ \\hline \\end{array}$\na) The table above gives the population of these two countries, in millions. Does country A or country B represent the population of Steadia? Enter just the letter of the country in the blank (A or B). [ANS]\nb) What is the approximate rate of change of the linear function? [ANS]\nc) What does the rate of change in your answer to (b) represent in practical terms? (select all that apply) [ANS] A. The total amount (in millions) the population grows from 1950 to 2000. B. The percent the population grows each year. C. The amount (in millions) the population grows every 10 years. D. The amount (in millions) the population grows each year. E. The number of years it takes for the population to increase by one million people. F. None of the above.\nd) Estimate the population of Steadia in 1987. [ANS] million people", "answer_v3": [ "B", "0.18", "D", "15.56" ], "answer_type_v3": [ "MCS", "NV", "MCS", "NV" ], "options_v3": [ [ "A", "B" ], [], [ "A", "B", "C", "D", "E", "F" ], [] ] }, { "id": "Algebra_0260", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "lines", "linear functions", "slope-intercept", "point-slope form" ], "problem_v1": "A theater manager graphed weekly profits as a function of the number of patrons and found that the relationship was linear. One week profit was \\$13,408 when 1328 patrons attended. Another week 1548 patrons produced a profit of \\$15,828.\n(a) Find a formula for weekly profit, $y$, as a function of the number of patrons, $x$. Do not enter any commas in your formula. $y=$ [ANS]\n(b) How much will profits increase if 1 more patron goes to the theater? \\$ [ANS] (no comma in your answer) (c) What number of patrons is closest to the break-even point (that is, the number of patrons for which there is as close to zero profit as possible)? number=[ANS] patrons (no comma in your answer) d) Find a formula for the number of patrons as a function of the profit. Do not enter any commas in your formula. $x=\\ $ [ANS]\nf) If the weekly profit was \\$20,338, how many patrons attended the theater? number=[ANS] patrons (no comma in your answer)", "answer_v1": [ "11*x+-1200", "11", "109", "0.0909091*y+109.091", "1958" ], "answer_type_v1": [ "EX", "NV", "NV", "EX", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "A theater manager graphed weekly profits as a function of the number of patrons and found that the relationship was linear. One week profit was \\$8,148 when 1314 patrons attended. Another week 1494 patrons produced a profit of \\$9,408.\n(a) Find a formula for weekly profit, $y$, as a function of the number of patrons, $x$. Do not enter any commas in your formula. $y=$ [ANS]\n(b) How much will profits increase if 1 more patron goes to the theater? \\$ [ANS] (no comma in your answer) (c) What number of patrons is closest to the break-even point (that is, the number of patrons for which there is as close to zero profit as possible)? number=[ANS] patrons (no comma in your answer) d) Find a formula for the number of patrons as a function of the profit. Do not enter any commas in your formula. $x=\\ $ [ANS]\nf) If the weekly profit was \\$13,188, how many patrons attended the theater? number=[ANS] patrons (no comma in your answer)", "answer_v2": [ "7*x+-1050", "7", "150", "0.142857*y+150", "2034" ], "answer_type_v2": [ "EX", "NV", "NV", "EX", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "A theater manager graphed weekly profits as a function of the number of patrons and found that the relationship was linear. One week profit was \\$10,003 when 1318 patrons attended. Another week 1528 patrons produced a profit of \\$11,788.\n(a) Find a formula for weekly profit, $y$, as a function of the number of patrons, $x$. Do not enter any commas in your formula. $y=$ [ANS]\n(b) How much will profits increase if 1 more patron goes to the theater? \\$ [ANS] (no comma in your answer) (c) What number of patrons is closest to the break-even point (that is, the number of patrons for which there is as close to zero profit as possible)? number=[ANS] patrons (no comma in your answer) d) Find a formula for the number of patrons as a function of the profit. Do not enter any commas in your formula. $x=\\ $ [ANS]\nf) If the weekly profit was \\$15,103, how many patrons attended the theater? number=[ANS] patrons (no comma in your answer)", "answer_v3": [ "8.5*x+-1200", "8.5", "141", "0.117647*y+141.176", "1918" ], "answer_type_v3": [ "EX", "NV", "NV", "EX", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Algebra_0261", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "lines", "parallel", "slope-intercept", "perpendicular" ], "problem_v1": "The cost of a Frigbox refrigerator is \\$850, and depreciates \\$50 each year. The cost of an Arctic Air refrigerator is \\$1480, and it depreciates \\$140 per year.\n(a) Find an equation for the value of the Frigbox, $F$, $t$ years after it is purchased. $F(t)=$ [ANS]\n(b) Find an equation for the value of the Arctic Air, $A$, $t$ years after it is purchased. $A(t)=$ [ANS]\n(c) If a Frigbox and an Arctic Air are bought at the same time, when do the two refrigerators have equal value? In [ANS] years.", "answer_v1": [ "850-50*t", "1480-140*t", "7" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The cost of a Frigbox refrigerator is \\$900, and depreciates \\$40 each year. The cost of an Arctic Air refrigerator is \\$1220, and it depreciates \\$120 per year.\n(a) Find an equation for the value of the Frigbox, $F$, $t$ years after it is purchased. $F(t)=$ [ANS]\n(b) Find an equation for the value of the Arctic Air, $A$, $t$ years after it is purchased. $A(t)=$ [ANS]\n(c) If a Frigbox and an Arctic Air are bought at the same time, when do the two refrigerators have equal value? In [ANS] years.", "answer_v2": [ "900-40*t", "1220-120*t", "4" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The cost of a Frigbox refrigerator is \\$875, and depreciates \\$40 each year. The cost of an Arctic Air refrigerator is \\$1325, and it depreciates \\$130 per year.\n(a) Find an equation for the value of the Frigbox, $F$, $t$ years after it is purchased. $F(t)=$ [ANS]\n(b) Find an equation for the value of the Arctic Air, $A$, $t$ years after it is purchased. $A(t)=$ [ANS]\n(c) If a Frigbox and an Arctic Air are bought at the same time, when do the two refrigerators have equal value? In [ANS] years.", "answer_v3": [ "875-40*t", "1325-130*t", "5" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0262", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "algebra", "system of equations", "non-linear" ], "problem_v1": "Charlie is trying to allocate his study time this weekend. He can spend time working with either his math tutor or his chemistry tutor to prepare for next week's tests. His math tutor charges \\$ 30 per hour. His chemistry tutor charges \\$ 50 per hour. He has \\$ 250 to spend on tutoring, but each hour with the math tutor requires 4 aspirin and 1 hours of sleep to recover. Each hour with the chemistry tutor requires 2 aspirin and 4 hour of sleep to recover. Charlie has only 30 aspirin left, and can only afford to sleep for 16 hours this weekend. If each hour of math tutoring will improve his grade by 3 points and each hour of chemistry tutoring will improve his grade by 4 points, how many hours should he spend with each tutor in order to improve his grades the most?\nCharlie should spend [ANS] hours with his math tutor and [ANS] hours with his chemistry tutor to improve his grades by a total of [ANS] points.", "answer_v1": [ "7.14285714285714", "0.714285714285714", "24.2857142857143" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Charlie is trying to allocate his study time this weekend. He can spend time working with either his math tutor or his chemistry tutor to prepare for next week's tests. His math tutor charges \\$ 20 per hour. His chemistry tutor charges \\$ 40 per hour. He has \\$ 240 to spend on tutoring, but each hour with the math tutor requires 3 aspirin and 1 hours of sleep to recover. Each hour with the chemistry tutor requires 1 aspirin and 3 hour of sleep to recover. Charlie has only 30 aspirin left, and can only afford to sleep for 15 hours this weekend. If each hour of math tutoring will improve his grade by 5 points and each hour of chemistry tutoring will improve his grade by 1 points, how many hours should he spend with each tutor in order to improve his grades the most?\nCharlie should spend [ANS] hours with his math tutor and [ANS] hours with his chemistry tutor to improve his grades by a total of [ANS] points.", "answer_v2": [ "10", "0", "50" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Charlie is trying to allocate his study time this weekend. He can spend time working with either his math tutor or his chemistry tutor to prepare for next week's tests. His math tutor charges \\$ 20 per hour. His chemistry tutor charges \\$ 30 per hour. He has \\$ 240 to spend on tutoring, but each hour with the math tutor requires 3 aspirin and 1 hours of sleep to recover. Each hour with the chemistry tutor requires 1 aspirin and 4 hour of sleep to recover. Charlie has only 30 aspirin left, and can only afford to sleep for 18 hours this weekend. If each hour of math tutoring will improve his grade by 4 points and each hour of chemistry tutoring will improve his grade by 2 points, how many hours should he spend with each tutor in order to improve his grades the most?\nCharlie should spend [ANS] hours with his math tutor and [ANS] hours with his chemistry tutor to improve his grades by a total of [ANS] points.", "answer_v3": [ "9.42857142857143", "1.71428571428571", "41.1428571428571" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0263", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "algebra", "pair of lines", "story question" ], "problem_v1": "The admission fee at an amusement park is 1.5 dollars for children and 4 dollars for adults. On a certain day, 336 people entered the park, and the admission fees collected totaled 904 dollars. How many children and how many adults were admitted? Your answer is number of children equals [ANS]\nnumber of adults equals [ANS]", "answer_v1": [ "176", "160" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The admission fee at an amusement park is 1.5 dollars for children and 4 dollars for adults. On a certain day, 303 people entered the park, and the admission fees collected totaled 942 dollars. How many children and how many adults were admitted? Your answer is number of children equals [ANS]\nnumber of adults equals [ANS]", "answer_v2": [ "108", "195" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The admission fee at an amusement park is 1.5 dollars for children and 4 dollars for adults. On a certain day, 292 people entered the park, and the admission fees collected totaled 838 dollars. How many children and how many adults were admitted? Your answer is number of children equals [ANS]\nnumber of adults equals [ANS]", "answer_v3": [ "132", "160" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0264", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "algebra", "pair of lines", "story question" ], "problem_v1": "A man has 31 coins in his pocket, all of which are dimes and quarters. If the total value of his change is 520 cents, how many dimes and how many quarters does he have? Your answer is number of dimes equals [ANS]\nnumber of quarters equals [ANS]", "answer_v1": [ "17", "14" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A man has 25 coins in his pocket, all of which are dimes and quarters. If the total value of his change is 535 cents, how many dimes and how many quarters does he have? Your answer is number of dimes equals [ANS]\nnumber of quarters equals [ANS]", "answer_v2": [ "6", "19" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A man has 24 coins in his pocket, all of which are dimes and quarters. If the total value of his change is 450 cents, how many dimes and how many quarters does he have? Your answer is number of dimes equals [ANS]\nnumber of quarters equals [ANS]", "answer_v3": [ "10", "14" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0265", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "Algebra", "Functions", "Modeling" ], "problem_v1": "A company that makes thing-a-ma-bobs has a start up cost of \\$40108. It costs the company \\$2.75 to make each thing-a-ma-bob and the company charges \\$4.86 for each thing-a-ma-bob. Let $x$ represent the number of thing-a-ma-bobs made.\nWrite the cost function for this company. $C(x)$=[ANS]\nWrite the revenue function for this company. $R(x)$=[ANS]\nWrite the profit function for this company. $P(x)=R(x)-C(x)$=[ANS]\nWhat is the minimum number of thing-a-ma-bobs that the company must pruduce and sell to make a profit? Answer=[ANS]", "answer_v1": [ "2.75 x + 40108", "4.86 x", "4.86 x -(2.75 x + 40108)", "19009" ], "answer_type_v1": [ "EX", "EX", "EX", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "A company that makes thing-a-ma-bobs has a start up cost of \\$13320. It costs the company \\$3.8 to make each thing-a-ma-bob and the company charges \\$4 for each thing-a-ma-bob. Let $x$ represent the number of thing-a-ma-bobs made.\nWrite the cost function for this company. $C(x)$=[ANS]\nWrite the revenue function for this company. $R(x)$=[ANS]\nWrite the profit function for this company. $P(x)=R(x)-C(x)$=[ANS]\nWhat is the minimum number of thing-a-ma-bobs that the company must pruduce and sell to make a profit? Answer=[ANS]", "answer_v2": [ "3.8 x + 13320", "4 x", "4 x -(3.8 x + 13320)", "66601" ], "answer_type_v2": [ "EX", "EX", "EX", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "A company that makes thing-a-ma-bobs has a start up cost of \\$22538. It costs the company \\$2.82 to make each thing-a-ma-bob and the company charges \\$3.83 for each thing-a-ma-bob. Let $x$ represent the number of thing-a-ma-bobs made.\nWrite the cost function for this company. $C(x)$=[ANS]\nWrite the revenue function for this company. $R(x)$=[ANS]\nWrite the profit function for this company. $P(x)=R(x)-C(x)$=[ANS]\nWhat is the minimum number of thing-a-ma-bobs that the company must pruduce and sell to make a profit? Answer=[ANS]", "answer_v3": [ "2.82 x + 22538", "3.83 x", "3.83 x -(2.82 x + 22538)", "22315" ], "answer_type_v3": [ "EX", "EX", "EX", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Algebra_0266", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "Algebra", "Modeling", "word problem", "percentage" ], "problem_v1": "What quantity of 70 per cent acid solution must be mixed with a 25 solution to produce 756 mL of a 50 per cent solution? [ANS]", "answer_v1": [ "420" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "What quantity of 80 per cent acid solution must be mixed with a 20 solution to produce 120 mL of a 50 per cent solution? [ANS]", "answer_v2": [ "60" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "What quantity of 75 per cent acid solution must be mixed with a 20 solution to produce 330 mL of a 50 per cent solution? [ANS]", "answer_v3": [ "180" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0267", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "Algebra", "Modeling", "algebra", "percent" ], "problem_v1": "$250$ is $50$ percent of what number?\nYour answers is: [ANS]", "answer_v1": [ "500" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "$110$ is $75$ percent of what number?\nYour answers is: [ANS]", "answer_v2": [ "146.666666666667" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "$160$ is $55$ percent of what number?\nYour answers is: [ANS]", "answer_v3": [ "290.909090909091" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0268", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "2", "keywords": [ "Algebra", "Modeling", "word problem' 'cutting wire", "quadratic" ], "problem_v1": "Mutt and Jeff need to paint a fence. Mutt can do the job alone 7 hours faster than Jeff. If together they work for 27 hours and finish only $\\frac{5}{6}$ of the job, how long would Jeff need to do the job alone?\nYour answer must be a number. No arithmetic operations are allowed.\nIt would take Jeff [ANS] hours to do the job alone.", "answer_v1": [ "68.4884949023425" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Mutt and Jeff need to paint a fence. Mutt can do the job alone 2 hours faster than Jeff. If together they work for 38 hours and finish only $\\frac{2}{3}$ of the job, how long would Jeff need to do the job alone?\nYour answer must be a number. No arithmetic operations are allowed.\nIt would take Jeff [ANS] hours to do the job alone.", "answer_v2": [ "115.008771254957" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Mutt and Jeff need to paint a fence. Mutt can do the job alone 4 hours faster than Jeff. If together they work for 27 hours and finish only $\\frac{2}{3}$ of the job, how long would Jeff need to do the job alone?\nYour answer must be a number. No arithmetic operations are allowed.\nIt would take Jeff [ANS] hours to do the job alone.", "answer_v3": [ "83.0493526458808" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0269", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "Algebra", "Modeling", "word problem", "percent" ], "problem_v1": "The radiator in a car is filled with a solution of 75 per cent antifreeze and 25 per cent water. The manufacturer of the antifreeze suggests that for summer driving, optimal cooling of the engine is obtained with only 50 per cent antifreeze. If the capacity of the raditor is 4.2 liters, how much coolant (in liters) must be drained and replaced with pure water to reduce the antifreeze concentration to 50 per cent? [ANS]", "answer_v1": [ "1.4" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The radiator in a car is filled with a solution of 60 per cent antifreeze and 40 per cent water. The manufacturer of the antifreeze suggests that for summer driving, optimal cooling of the engine is obtained with only 50 per cent antifreeze. If the capacity of the raditor is 4.9 liters, how much coolant (in liters) must be drained and replaced with pure water to reduce the antifreeze concentration to 50 per cent? [ANS]", "answer_v2": [ "0.816666666666667" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The radiator in a car is filled with a solution of 65 per cent antifreeze and 35 per cent water. The manufacturer of the antifreeze suggests that for summer driving, optimal cooling of the engine is obtained with only 50 per cent antifreeze. If the capacity of the raditor is 4.2 liters, how much coolant (in liters) must be drained and replaced with pure water to reduce the antifreeze concentration to 50 per cent? [ANS]", "answer_v3": [ "0.969230769230769" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0270", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "2", "keywords": [ "Algebra", "Modeling", "algebra", "application of linear equation" ], "problem_v1": "This exercise concerns with modeling with linear equations.\nOne positive number is one-fifth of another number. The difference between the two numbers is 272, find the numbers.\nThe two numbers in increasing order are [ANS] and [ANS]", "answer_v1": [ "68", "340" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "This exercise concerns with modeling with linear equations.\nOne positive number is one-fifth of another number. The difference between the two numbers is 100, find the numbers.\nThe two numbers in increasing order are [ANS] and [ANS]", "answer_v2": [ "25", "125" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "This exercise concerns with modeling with linear equations.\nOne positive number is one-fifth of another number. The difference between the two numbers is 160, find the numbers.\nThe two numbers in increasing order are [ANS] and [ANS]", "answer_v3": [ "40", "200" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0271", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "2", "keywords": [ "Algebra", "Modeling", "algebra", "application of linear equations" ], "problem_v1": "A rectangular room is $1.7$ times as long as it is wide, and its perimeter is $33$ meters. Find the dimension of the room.\nThe length is: [ANS] meters and the width is [ANS] meters.", "answer_v1": [ "10.3888888888889", "6.11111111111111" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A rectangular room is $2$ times as long as it is wide, and its perimeter is $25$ meters. Find the dimension of the room.\nThe length is: [ANS] meters and the width is [ANS] meters.", "answer_v2": [ "8.33333333333333", "4.16666666666667" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A rectangular room is $1.7$ times as long as it is wide, and its perimeter is $28$ meters. Find the dimension of the room.\nThe length is: [ANS] meters and the width is [ANS] meters.", "answer_v3": [ "8.81481481481481", "5.18518518518519" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0272", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "Algebra", "Modeling", "word problem" ], "problem_v1": "Two cyclists, 108 miles apart, start riding toward each other at the same time. One cycles 3 times as fast as the other. If they meet 3 hours later, what is the speed (in mi/h) of the faster cyclist? [ANS]", "answer_v1": [ "27" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Two cyclists, 40 miles apart, start riding toward each other at the same time. One cycles 3 times as fast as the other. If they meet 2 hours later, what is the speed (in mi/h) of the faster cyclist? [ANS]", "answer_v2": [ "15" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Two cyclists, 48 miles apart, start riding toward each other at the same time. One cycles 3 times as fast as the other. If they meet 2 hours later, what is the speed (in mi/h) of the faster cyclist? [ANS]", "answer_v3": [ "18" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0273", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "", "keywords": [ "Algebra", "Modeling", "word problem" ], "problem_v1": "After robbing a bank in Dodge City, a robber gallops off at 13 mi/h. 30 minutes later, the marshall leaves to pursue the robber at 16 mi/h. How long (in hours) does it take the marshall to catch up to the robber? [ANS]", "answer_v1": [ "2.16666666666667" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "After robbing a bank in Dodge City, a robber gallops off at 14 mi/h. 10 minutes later, the marshall leaves to pursue the robber at 15 mi/h. How long (in hours) does it take the marshall to catch up to the robber? [ANS]", "answer_v2": [ "2.33333333333333" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "After robbing a bank in Dodge City, a robber gallops off at 13 mi/h. 10 minutes later, the marshall leaves to pursue the robber at 15 mi/h. How long (in hours) does it take the marshall to catch up to the robber? [ANS]", "answer_v3": [ "1.08333333333333" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0274", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "algebra", "modeling", "equations" ], "problem_v1": "Wilma drove at an average speed of $55$ mi/h from her home in City A to visit her sister in City B. She stayed in City B $20$ hours, and on the trip back averaged $50$ mi/h. She returned home $47$ hours after leaving. How many miles is City A from City B\nYour answer is: [ANS]", "answer_v1": [ "707.142857142857" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Wilma drove at an average speed of $35$ mi/h from her home in City A to visit her sister in City B. She stayed in City B $10$ hours, and on the trip back averaged $60$ mi/h. She returned home $43$ hours after leaving. How many miles is City A from City B\nYour answer is: [ANS]", "answer_v2": [ "729.473684210526" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Wilma drove at an average speed of $40$ mi/h from her home in City A to visit her sister in City B. She stayed in City B $15$ hours, and on the trip back averaged $50$ mi/h. She returned home $46$ hours after leaving. How many miles is City A from City B\nYour answer is: [ANS]", "answer_v3": [ "688.888888888889" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0275", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "2", "keywords": [ "Algebra", "Modeling", "algebra", "percent" ], "problem_v1": "Your weekly paycheck is $25$ percent less than your coworker's. Your two paychecks total $775$. Find the amount of each paycheck.\nYour coworker's is: [ANS] and yours is [ANS].", "answer_v1": [ "442.857142857143", "332.142857142857" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Your weekly paycheck is $10$ percent less than your coworker's. Your two paychecks total $880$. Find the amount of each paycheck.\nYour coworker's is: [ANS] and yours is [ANS].", "answer_v2": [ "463.157894736842", "416.842105263158" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Your weekly paycheck is $15$ percent less than your coworker's. Your two paychecks total $780$. Find the amount of each paycheck.\nYour coworker's is: [ANS] and yours is [ANS].", "answer_v3": [ "421.621621621622", "358.378378378378" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0276", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "function", "modeling" ], "problem_v1": "Biologists have noticed that the chirping of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 118 chirps per minute at 71 degrees Fahrenheit and 176 chirps per minute at 87 degrees Fahrenheit.\nFind a linear equation that models the temperature $T$ as a function of the number of chirps per minute $N$. $T(N)=$ [ANS]\nIf the crickets are chirping at 153 chirps per minute, estimate the temperature: Temperature=[ANS]", "answer_v1": [ "((87 - 71)/(176 - 118))*(N - 176) + 87", "80.6551724137931" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Biologists have noticed that the chirping of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 110 chirps per minute at 75 degrees Fahrenheit and 171 chirps per minute at 83 degrees Fahrenheit.\nFind a linear equation that models the temperature $T$ as a function of the number of chirps per minute $N$. $T(N)=$ [ANS]\nIf the crickets are chirping at 160 chirps per minute, estimate the temperature: Temperature=[ANS]", "answer_v2": [ "((83 - 75)/(171 - 110))*(N - 171) + 83", "81.5573770491803" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Biologists have noticed that the chirping of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 113 chirps per minute at 71 degrees Fahrenheit and 173 chirps per minute at 86 degrees Fahrenheit.\nFind a linear equation that models the temperature $T$ as a function of the number of chirps per minute $N$. $T(N)=$ [ANS]\nIf the crickets are chirping at 152 chirps per minute, estimate the temperature: Temperature=[ANS]", "answer_v3": [ "((86 - 71)/(173 - 113))*(N - 173) + 86", "80.75" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0277", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "Equation", "Similar Triangle" ], "problem_v1": "At 3:00 PM a man 145 cm tall casts a shadow 142 cm long. At the same time, a tall building nearby casts a shadow 185 m long. How tall is the building? [ANS]\nGive your answer in meters. (You may need the fact that 100 cm=1 m.)", "answer_v1": [ "188.908450704225" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "At 3:00 PM a man 131 cm tall casts a shadow 149 cm long. At the same time, a tall building nearby casts a shadow 166 m long. How tall is the building? [ANS]\nGive your answer in meters. (You may need the fact that 100 cm=1 m.)", "answer_v2": [ "145.946308724832" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "At 3:00 PM a man 136 cm tall casts a shadow 142 cm long. At the same time, a tall building nearby casts a shadow 171 m long. How tall is the building? [ANS]\nGive your answer in meters. (You may need the fact that 100 cm=1 m.)", "answer_v3": [ "163.774647887324" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0278", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "algebra", "pair of lines", "story question" ], "problem_v1": "A man invests his savings in two accounts, one paying 6 percent and the other paying 10 percent simple interest per year. He puts twice as much in the lower-yielding account because it is less risky. His annual interest is 572 dollars. How much did he invest at each rate? Your answer is total in the account paying 6 percent interest is [ANS]\ntotal in the account paying 10 percent interest is [ANS]", "answer_v1": [ "5200", "2600" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A man invests his savings in two accounts, one paying 6 percent and the other paying 10 percent simple interest per year. He puts twice as much in the lower-yielding account because it is less risky. His annual interest is 2024 dollars. How much did he invest at each rate? Your answer is total in the account paying 6 percent interest is [ANS]\ntotal in the account paying 10 percent interest is [ANS]", "answer_v2": [ "18400", "9200" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A man invests his savings in two accounts, one paying 6 percent and the other paying 10 percent simple interest per year. He puts twice as much in the lower-yielding account because it is less risky. His annual interest is 1518 dollars. How much did he invest at each rate? Your answer is total in the account paying 6 percent interest is [ANS]\ntotal in the account paying 10 percent interest is [ANS]", "answer_v3": [ "13800", "6900" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0279", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "supply and demand''linear equations" ], "problem_v1": "Find the point of equilibrium for the following supply and demand equations where $x$ is number of units and $p$ is the price per unit. Demand: $p=231-0.000060x$ Supply: $p=145+0.000370x$ Number of units for equilibrium=[ANS]\nPrice per unit at equilibrium=[ANS]", "answer_v1": [ "200000", "219" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the point of equilibrium for the following supply and demand equations where $x$ is number of units and $p$ is the price per unit. Demand: $p=40-0.000100x$ Supply: $p=13+0.000170x$ Number of units for equilibrium=[ANS]\nPrice per unit at equilibrium=[ANS]", "answer_v2": [ "100000", "30" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the point of equilibrium for the following supply and demand equations where $x$ is number of units and $p$ is the price per unit. Demand: $p=105-0.000070x$ Supply: $p=35+0.000280x$ Number of units for equilibrium=[ANS]\nPrice per unit at equilibrium=[ANS]", "answer_v3": [ "200000", "91" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0280", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "secant lines", "tangent line" ], "problem_v1": "At the surface of the ocean, the water pressure is the same as the air pressure above the water, about $15 \\quad lb/in^2$, Below the surface the water pressure increases by about $4.84 \\quad lb/in^2$ for every 10 ft of descent. Write a function $f(x)$ which expresses the water pressure in pounds per square inch as a function of the depth in inches below the ocean surface. $f(x)=$ [ANS]\nAt what depth is the pressure $100 \\quad lb/in^2$? [ANS] in", "answer_v1": [ "0.0403333333333333*x +15", "2107.43801652893" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "At the surface of the ocean, the water pressure is the same as the air pressure above the water, about $15 \\quad lb/in^2$, Below the surface the water pressure increases by about $3.44 \\quad lb/in^2$ for every 10 ft of descent. Write a function $f(x)$ which expresses the water pressure in pounds per square inch as a function of the depth in inches below the ocean surface. $f(x)=$ [ANS]\nAt what depth is the pressure $120 \\quad lb/in^2$? [ANS] in", "answer_v2": [ "0.0286666666666667*x +15", "3662.79069767442" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "At the surface of the ocean, the water pressure is the same as the air pressure above the water, about $15 \\quad lb/in^2$, Below the surface the water pressure increases by about $3.94 \\quad lb/in^2$ for every 10 ft of descent. Write a function $f(x)$ which expresses the water pressure in pounds per square inch as a function of the depth in inches below the ocean surface. $f(x)=$ [ANS]\nAt what depth is the pressure $100 \\quad lb/in^2$? [ANS] in", "answer_v3": [ "0.0328333333333333*x +15", "2588.83248730964" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0281", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "calculus", "derivative" ], "problem_v1": "The monthly charge for a waste collection service is 1265 dollars for 100 kg of waste and 1985 dollars for 160 kg of waste.\n(a) Find a linear model for the cost, $C$, of waste collection as a function of the number of kilograms, $w$. $C=$ [ANS]\n(b) What is the slope of the line found in part (a)? Slope=[ANS]\nThink about the interpretation of the slope: are the units of the slope [ANS] A. dollars per kilogram B. kilograms per dollar C. kilograms D. dollars\n(c) What is the value of the vertical intercept of the line found in part (a)? Value=[ANS]\nThink about the interpretation of the intercept: are the units of the intercept [ANS] A. kilograms per dollar B. kilograms C. dollars per kilogram D. dollars", "answer_v1": [ "12*w+65", "12", "A", "65", "D" ], "answer_type_v1": [ "EX", "NV", "MCS", "NV", "MCS" ], "options_v1": [ [], [], [ "A", "B", "C", "D" ], [], [ "A", "B", "C", "D" ] ], "problem_v2": "The monthly charge for a waste collection service is 2025 dollars for 100 kg of waste and 2525 dollars for 125 kg of waste.\n(a) Find a linear model for the cost, $C$, of waste collection as a function of the number of kilograms, $w$. $C=$ [ANS]\n(b) What is the slope of the line found in part (a)? Slope=[ANS]\nThink about the interpretation of the slope: are the units of the slope [ANS] A. kilograms B. kilograms per dollar C. dollars D. dollars per kilogram\n(c) What is the value of the vertical intercept of the line found in part (a)? Value=[ANS]\nThink about the interpretation of the intercept: are the units of the intercept [ANS] A. dollars B. dollars per kilogram C. kilograms per dollar D. kilograms", "answer_v2": [ "20*w+25", "20", "D", "25", "A" ], "answer_type_v2": [ "EX", "NV", "MCS", "NV", "MCS" ], "options_v2": [ [], [], [ "A", "B", "C", "D" ], [], [ "A", "B", "C", "D" ] ], "problem_v3": "The monthly charge for a waste collection service is 1440 dollars for 100 kg of waste and 1930 dollars for 135 kg of waste.\n(a) Find a linear model for the cost, $C$, of waste collection as a function of the number of kilograms, $w$. $C=$ [ANS]\n(b) What is the slope of the line found in part (a)? Slope=[ANS]\nThink about the interpretation of the slope: are the units of the slope [ANS] A. dollars per kilogram B. dollars C. kilograms D. kilograms per dollar\n(c) What is the value of the vertical intercept of the line found in part (a)? Value=[ANS]\nThink about the interpretation of the intercept: are the units of the intercept [ANS] A. dollars B. dollars per kilogram C. kilograms D. kilograms per dollar", "answer_v3": [ "14*w+40", "14", "A", "40", "A" ], "answer_type_v3": [ "EX", "NV", "MCS", "NV", "MCS" ], "options_v3": [ [], [], [ "A", "B", "C", "D" ], [], [ "A", "B", "C", "D" ] ] }, { "id": "Algebra_0282", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "average rate of change" ], "problem_v1": "Suppose that you have zero dollars now and that the average rate of change in your net worth is \\$8000 per year. How much money will you have in 50 years? answer=[ANS] dollars", "answer_v1": [ "400000" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that you have zero dollars now and that the average rate of change in your net worth is \\$3000 per year. How much money will you have in 70 years? answer=[ANS] dollars", "answer_v2": [ "210000" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that you have zero dollars now and that the average rate of change in your net worth is \\$5000 per year. How much money will you have in 50 years? answer=[ANS] dollars", "answer_v3": [ "250000" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0283", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "", "keywords": [ "system of equations" ], "problem_v1": "One day, a deli sold a total of 43 hamburgers and hot dogs. The revenue from these sales was \\$186.21. The hamburgers were \\$5.07 each and the hot dogs cost \\$2.95 each. a) Write a mathematical expression for the number of burgers, $x$, and the number of hot dogs, $y$, sold that day. Answer: [ANS]\nb) Write a mathematical expression for the amount of revenue from this day's sales. Use $x$ for the number of burgers and $y$ for the number of hot dogs. Answer: [ANS]\nc) How many burgers did the store sell that day? Answer: [ANS]", "answer_v1": [ "x + y = 43", "5.07 * x + 2.95 * y = 186.21", "28" ], "answer_type_v1": [ "EQ", "EQ", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "One day, a deli sold a total of 39 hamburgers and hot dogs. The revenue from these sales was \\$113.35. The hamburgers were \\$3.91 each and the hot dogs cost \\$1.85 each. a) Write a mathematical expression for the number of burgers, $x$, and the number of hot dogs, $y$, sold that day. Answer: [ANS]\nb) Write a mathematical expression for the amount of revenue from this day's sales. Use $x$ for the number of burgers and $y$ for the number of hot dogs. Answer: [ANS]\nc) How many hot dogs did the store sell that day? Answer: [ANS]", "answer_v2": [ "x + y = 39", "3.91 * x + 1.85 * y = 113.35", "19" ], "answer_type_v2": [ "EQ", "EQ", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "One day, a deli sold a total of 39 hamburgers and hot dogs. The revenue from these sales was \\$136.49. The hamburgers were \\$4.23 each and the hot dogs cost \\$2.45 each. a) Write a mathematical expression for the number of burgers, $x$, and the number of hot dogs, $y$, sold that day. Answer: [ANS]\nb) Write a mathematical expression for the amount of revenue from this day's sales. Use $x$ for the number of burgers and $y$ for the number of hot dogs. Answer: [ANS]\nc) How many burgers did the store sell that day? Answer: [ANS]", "answer_v3": [ "x + y = 39", "4.23 * x + 2.45 * y = 136.49", "23" ], "answer_type_v3": [ "EQ", "EQ", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0284", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "2", "keywords": [ "slope", "y-intercept" ], "problem_v1": "The linear function $y=-0.3x+39$ models the percentage of men in a certain city, $y$, smoking cigarettes $x$ years after 1995. Find the slope for the model. Then describe what this means in terms of the rate of change of male smokers in this city over time. a) $m=$ [ANS]\nb) The percentage of men in this city who smoke has [ANS] (Enter: increased or decreased or not changed) at a rate of [ANS] \\% per year after 1995.", "answer_v1": [ "-0.3", "decreased", "0.3" ], "answer_type_v1": [ "NV", "MCS", "NV" ], "options_v1": [ [], [ "increased", "decreased" ], [] ], "problem_v2": "The linear function $y=-0.6x+48$ models the percentage of men in a certain city, $y$, smoking cigarettes $x$ years after 1985. Find the slope for the model. Then describe what this means in terms of the rate of change of male smokers in this city over time. a) $m=$ [ANS]\nb) The percentage of men in this city who smoke has [ANS] (Enter: increased or decreased or not changed) at a rate of [ANS] \\% per year after 1985.", "answer_v2": [ "-0.6", "decreased", "0.6" ], "answer_type_v2": [ "NV", "MCS", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The linear function $y=-0.5x+39$ models the percentage of men in a certain city, $y$, smoking cigarettes $x$ years after 1990. Find the slope for the model. Then describe what this means in terms of the rate of change of male smokers in this city over time. a) $m=$ [ANS]\nb) The percentage of men in this city who smoke has [ANS] (Enter: increased or decreased or not changed) at a rate of [ANS] \\% per year after 1990.", "answer_v3": [ "-0.5", "decreased", "0.5" ], "answer_type_v3": [ "NV", "MCS", "NV" ], "options_v3": [ [], [ "increased", "decreased" ], [] ] }, { "id": "Algebra_0285", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "mathematical models", "problem solving", "formulas", "algebra" ], "problem_v1": "Video Store A charges \\$10 to rent a video game for one week. Although only members can rent from the store, membership is free. Video Store B charges only \\$3 to rent a video game for one week. Only members can rent from the store and membership is \\$168 per year. After how many video game rentals will the total amount spent at each store be the same? a) Write an equation to model the problem. Let x represent the number of video game rentals. Answer: [ANS]\nb) Solve your equation to answer the question. (Note: Your answer should be in the form 29 rentals. 29 rentals.) Answer: [ANS]", "answer_v1": [ "10 * x = 3 * x + 168", "24" ], "answer_type_v1": [ "EQ", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Video Store A charges \\$7 to rent a video game for one week. Although only members can rent from the store, membership is free. Video Store B charges only \\$4 to rent a video game for one week. Only members can rent from the store and membership is \\$51 per year. After how many video game rentals will the total amount spent at each store be the same? a) Write an equation to model the problem. Let x represent the number of video game rentals. Answer: [ANS]\nb) Solve your equation to answer the question. (Note: Your answer should be in the form 29 rentals. 29 rentals.) Answer: [ANS]", "answer_v2": [ "7 * x = 4 * x + 51", "17" ], "answer_type_v2": [ "EQ", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Video Store A charges \\$8 to rent a video game for one week. Although only members can rent from the store, membership is free. Video Store B charges only \\$3 to rent a video game for one week. Only members can rent from the store and membership is \\$95 per year. After how many video game rentals will the total amount spent at each store be the same? a) Write an equation to model the problem. Let x represent the number of video game rentals. Answer: [ANS]\nb) Solve your equation to answer the question. (Note: Your answer should be in the form 29 rentals. 29 rentals.) Answer: [ANS]", "answer_v3": [ "8 * x = 3 * x + 95", "19" ], "answer_type_v3": [ "EQ", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0286", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "mathematical models", "problem solving", "formulas", "algebra" ], "problem_v1": "60\\% of what number is 72? a) Write an equation to model the problem. Use x to represent the number. Answer: [ANS]\nb) Solve the equation to find the number: Answer: [ANS]", "answer_v1": [ "0.6 * x=72", "120" ], "answer_type_v1": [ "EQ", "NV" ], "options_v1": [ [], [] ], "problem_v2": "90\\% of what number is 18? a) Write an equation to model the problem. Use x to represent the number. Answer: [ANS]\nb) Solve the equation to find the number: Answer: [ANS]", "answer_v2": [ "0.9 * x=18", "20" ], "answer_type_v2": [ "EQ", "NV" ], "options_v2": [ [], [] ], "problem_v3": "60\\% of what number is 30? a) Write an equation to model the problem. Use x to represent the number. Answer: [ANS]\nb) Solve the equation to find the number: Answer: [ANS]", "answer_v3": [ "0.6 * x=30", "50" ], "answer_type_v3": [ "EQ", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0287", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "mathematical models", "problem solving", "formulas", "algebra" ], "problem_v1": "The selling price of a textbook is \\$144.00. If the markup is 20\\% of the bookstore's cost, what is the bookstore's cost of the textbook? a) Write an equation to model the problem. Use x to represent the number. Answer: [ANS]\nb) Solve the equation to find the bookstore's cost. (Note: Your answer should be in the form \\$ddd.cc. \\$ddd.cc.) Answer: [ANS]", "answer_v1": [ "x + 0.2 * x = 144", "120.00" ], "answer_type_v1": [ "EQ", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The selling price of a textbook is \\$116.25. If the markup is 25\\% of the bookstore's cost, what is the bookstore's cost of the textbook? a) Write an equation to model the problem. Use x to represent the number. Answer: [ANS]\nb) Solve the equation to find the bookstore's cost. (Note: Your answer should be in the form \\$ddd.cc. \\$ddd.cc.) Answer: [ANS]", "answer_v2": [ "x + 0.25 * x = 116.25", "93.00" ], "answer_type_v2": [ "EQ", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The selling price of a textbook is \\$122.40. If the markup is 20\\% of the bookstore's cost, what is the bookstore's cost of the textbook? a) Write an equation to model the problem. Use x to represent the number. Answer: [ANS]\nb) Solve the equation to find the bookstore's cost. (Note: Your answer should be in the form \\$ddd.cc. \\$ddd.cc.) Answer: [ANS]", "answer_v3": [ "x + 0.2 * x = 122.4", "102.00" ], "answer_type_v3": [ "EQ", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0288", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "mathematical models", "problem solving", "formulas", "algebra" ], "problem_v1": "Including 8\\% sales tax, an inn charges \\$216.00 per night. Find the nightly cost at the inn before tax is added. a) Write an equation to model the problem. Let x represent the nightly cost before tax. Answer: [ANS]\nb) Solve the equation to find nightly cost before tax. (Note: Your answer should be in the form \\$ddd.cc. \\$ddd.cc.) Answer: [ANS]", "answer_v1": [ "x + 0.08 * x = 216 ", "200.00" ], "answer_type_v1": [ "EQ", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Including 5\\% sales tax, an inn charges \\$262.50 per night. Find the nightly cost at the inn before tax is added. a) Write an equation to model the problem. Let x represent the nightly cost before tax. Answer: [ANS]\nb) Solve the equation to find nightly cost before tax. (Note: Your answer should be in the form \\$ddd.cc. \\$ddd.cc.) Answer: [ANS]", "answer_v2": [ "x + 0.05 * x = 262.5 ", "250.00" ], "answer_type_v2": [ "EQ", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Including 5\\% sales tax, an inn charges \\$210.00 per night. Find the nightly cost at the inn before tax is added. a) Write an equation to model the problem. Let x represent the nightly cost before tax. Answer: [ANS]\nb) Solve the equation to find nightly cost before tax. (Note: Your answer should be in the form \\$ddd.cc. \\$ddd.cc.) Answer: [ANS]", "answer_v3": [ "x + 0.05 * x = 210 ", "200.00" ], "answer_type_v3": [ "EQ", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0289", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "mathematical models", "problem solving", "formulas", "algebra" ], "problem_v1": "After a 25\\% reduction, you purchase a television for \\$757.5. What as the television's price before the reduction? a) Write an equation to model the problem. Let x represent the price before the reduction. Answer: [ANS]\nb) Solve the equation to find the price before the reduction. (Note: Your answer should be in the form \\$ddd.cc. \\$ddd.cc.) Answer: [ANS]", "answer_v1": [ "x - 0.25 * x = 757.5", "1010.00" ], "answer_type_v1": [ "EQ", "NV" ], "options_v1": [ [], [] ], "problem_v2": "After a 10\\% reduction, you purchase a television for \\$1134. What as the television's price before the reduction? a) Write an equation to model the problem. Let x represent the price before the reduction. Answer: [ANS]\nb) Solve the equation to find the price before the reduction. (Note: Your answer should be in the form \\$ddd.cc. \\$ddd.cc.) Answer: [ANS]", "answer_v2": [ "x - 0.1 * x = 1134", "1260.00" ], "answer_type_v2": [ "EQ", "NV" ], "options_v2": [ [], [] ], "problem_v3": "After a 15\\% reduction, you purchase a television for \\$867. What as the television's price before the reduction? a) Write an equation to model the problem. Let x represent the price before the reduction. Answer: [ANS]\nb) Solve the equation to find the price before the reduction. (Note: Your answer should be in the form \\$ddd.cc. \\$ddd.cc.) Answer: [ANS]", "answer_v3": [ "x - 0.15 * x = 867", "1020.00" ], "answer_type_v3": [ "EQ", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0290", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "mathematical models", "problem solving", "formulas", "algebra" ], "problem_v1": "According to one mathematical model, the average life expenctancy for American men born in 1900 was 55 years. Life expectancy has increased by about 0.2 year for each birth year after 1900. If this trend continues, for which birth year will the average life expentancy be 77 years? a) Write an equation to model the problem. Let $t$ represent the number of years after 1900 and let $n$ represent men's life expectancy at that time. For example $t=12$ and $n=57.4$ in the year 1912. Answer: [ANS]\nb) Solve the equation, then answer the question given above. (Note: You are asked for a year, not a value for $t$. Answer: [ANS]", "answer_v1": [ "0.2*t-n = -55", "2010" ], "answer_type_v1": [ "EQ", "NV" ], "options_v1": [ [], [] ], "problem_v2": "According to one mathematical model, the average life expenctancy for American men born in 1900 was 55 years. Life expectancy has increased by about 0.2 year for each birth year after 1900. If this trend continues, for which birth year will the average life expentancy be 71 years? a) Write an equation to model the problem. Let $t$ represent the number of years after 1900 and let $n$ represent men's life expectancy at that time. For example $t=12$ and $n=57.4$ in the year 1912. Answer: [ANS]\nb) Solve the equation, then answer the question given above. (Note: You are asked for a year, not a value for $t$. Answer: [ANS]", "answer_v2": [ "0.2*t-n = -55", "1980" ], "answer_type_v2": [ "EQ", "NV" ], "options_v2": [ [], [] ], "problem_v3": "According to one mathematical model, the average life expenctancy for American men born in 1900 was 55 years. Life expectancy has increased by about 0.2 year for each birth year after 1900. If this trend continues, for which birth year will the average life expentancy be 73 years? a) Write an equation to model the problem. Let $t$ represent the number of years after 1900 and let $n$ represent men's life expectancy at that time. For example $t=12$ and $n=57.4$ in the year 1912. Answer: [ANS]\nb) Solve the equation, then answer the question given above. (Note: You are asked for a year, not a value for $t$. Answer: [ANS]", "answer_v3": [ "0.2*t-n = -55", "1990" ], "answer_type_v3": [ "EQ", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0291", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "", "keywords": [ "inequalities" ], "problem_v1": "For a certain county, the percentage, $P$, of voters who used electronic voting systems, such as optical scans, in national elections can be modeled by the formula P=3x+20.9, where $x$ is the number of years after 2001. In which years will more than 53.9\\% of the county's voters use electronic systems? Note: Enter your answer as, Voting years after yyyy--do not put a period at the end of the phrase. Answer: [ANS]", "answer_v1": [ "VOTING YEARS AFTER 2012" ], "answer_type_v1": [ "OE" ], "options_v1": [ [] ], "problem_v2": "For a certain county, the percentage, $P$, of voters who used electronic voting systems, such as optical scans, in national elections can be modeled by the formula P=2.3x+15, where $x$ is the number of years after 2002. In which years will more than 24.2\\% of the county's voters use electronic systems? Note: Enter your answer as, Voting years after yyyy--do not put a period at the end of the phrase. Answer: [ANS]", "answer_v2": [ "VOTING YEARS AFTER 2006" ], "answer_type_v2": [ "OE" ], "options_v2": [ [] ], "problem_v3": "For a certain county, the percentage, $P$, of voters who used electronic voting systems, such as optical scans, in national elections can be modeled by the formula P=2.5x+18.2, where $x$ is the number of years after 2001. In which years will more than 35.7\\% of the county's voters use electronic systems? Note: Enter your answer as, Voting years after yyyy--do not put a period at the end of the phrase. Answer: [ANS]", "answer_v3": [ "VOTING YEARS AFTER 2008" ], "answer_type_v3": [ "OE" ], "options_v3": [ [] ] }, { "id": "Algebra_0292", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "linear equations", "solving equations" ], "problem_v1": "The average cost of tuition and fees at public four year colleges in the United States can be modeled by the formula T=165x+2771, where T represents the average cost of tuition and fees for the school year ending x years after 1996. Use the formula to determine the year when tuition and fees at public U.S. colleges average \\$5081. (Note: Give the year. For example, if you calculate that the answer is 9 years after 1996, then enter 2005 2005.) Answer: [ANS]", "answer_v1": [ "2010" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The average cost of tuition and fees at public four year colleges in the United States can be modeled by the formula T=165x+2771, where T represents the average cost of tuition and fees for the school year ending x years after 1996. Use the formula to determine the year when tuition and fees at public U.S. colleges average \\$4421. (Note: Give the year. For example, if you calculate that the answer is 9 years after 1996, then enter 2005 2005.) Answer: [ANS]", "answer_v2": [ "2006" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The average cost of tuition and fees at public four year colleges in the United States can be modeled by the formula T=165x+2771, where T represents the average cost of tuition and fees for the school year ending x years after 1996. Use the formula to determine the year when tuition and fees at public U.S. colleges average \\$4586. (Note: Give the year. For example, if you calculate that the answer is 9 years after 1996, then enter 2005 2005.) Answer: [ANS]", "answer_v3": [ "2007" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0293", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "algebra", "modeling", "equations" ], "problem_v1": "A plumber and his assistant work together to replace the pipes in an old house. The plumber charges 45 dollars per hour for his own labor and 25 dollars for his assistant's labor. The plumber works twice as long as his assistant on this job, and the labor charge on the final bill is 7015. How long did the plumber work on this job?\nYour answer is [ANS] hours.", "answer_v1": [ "122" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A plumber and his assistant work together to replace the pipes in an old house. The plumber charges 45 dollars per hour for his own labor and 25 dollars for his assistant's labor. The plumber works twice as long as his assistant on this job, and the labor charge on the final bill is 920. How long did the plumber work on this job?\nYour answer is [ANS] hours.", "answer_v2": [ "16" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A plumber and his assistant work together to replace the pipes in an old house. The plumber charges 45 dollars per hour for his own labor and 25 dollars for his assistant's labor. The plumber works twice as long as his assistant on this job, and the labor charge on the final bill is 2990. How long did the plumber work on this job?\nYour answer is [ANS] hours.", "answer_v3": [ "52" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0294", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "algebra", "modeling", "equations" ], "problem_v1": "An executive in an engineering firm earns a monthly salary plus a Christmas bonus of $7300$ dollars. If she earns a total of $92400$ dollars per year, what is her monthly salary in dollars?\nYour answer is: [ANS]", "answer_v1": [ "7091.66666666667" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "An executive in an engineering firm earns a monthly salary plus a Christmas bonus of $5300$ dollars. If she earns a total of $98700$ dollars per year, what is her monthly salary in dollars?\nYour answer is: [ANS]", "answer_v2": [ "7783.33333333333" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "An executive in an engineering firm earns a monthly salary plus a Christmas bonus of $6000$ dollars. If she earns a total of $92800$ dollars per year, what is her monthly salary in dollars?\nYour answer is: [ANS]", "answer_v3": [ "7233.33333333333" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0295", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "algebra", "modeling", "equations" ], "problem_v1": "Stan and Hilda can mow the lawn in $45$ min if they work together. If Hilda works twice as fast as Stan, how long would it take Stan to mow the lawn alone?\nGive your answer in munites here: [ANS]", "answer_v1": [ "135" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Stan and Hilda can mow the lawn in $20$ min if they work together. If Hilda works twice as fast as Stan, how long would it take Stan to mow the lawn alone?\nGive your answer in munites here: [ANS]", "answer_v2": [ "60" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Stan and Hilda can mow the lawn in $30$ min if they work together. If Hilda works twice as fast as Stan, how long would it take Stan to mow the lawn alone?\nGive your answer in munites here: [ANS]", "answer_v3": [ "90" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0296", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [], "problem_v1": "In a sequence of five integers the third integer is the sum of the previous two, the fourth integer is the sum of the previous three and the fifth integer is the sum of the previous four. If the sum of the five integers is 176 then the third integer is [ANS].", "answer_v1": [ "22" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "In a sequence of five integers the third integer is the sum of the previous two, the fourth integer is the sum of the previous three and the fifth integer is the sum of the previous four. If the sum of the five integers is 136 then the third integer is [ANS].", "answer_v2": [ "17" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "In a sequence of five integers the third integer is the sum of the previous two, the fourth integer is the sum of the previous three and the fifth integer is the sum of the previous four. If the sum of the five integers is 144 then the third integer is [ANS].", "answer_v3": [ "18" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0297", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "2", "keywords": [ "algebra", "lines" ], "problem_v1": "Suppose a mining company will supply $110000$ tons of ore per month if the price is $90$ dollars per ton but will supply $69500$ tons per month if the price is $20$ dollars per ton. Assuming the supply function is of the form $y=mx+b$, find the slope, $m$ and y-intercept, $b$ $m$: [ANS]\n$b$: [ANS]", "answer_v1": [ "0.0017283950617284", "-100.123456790123" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose a mining company will supply $83000$ tons of ore per month if the price is $100$ dollars per ton but will supply $62000$ tons per month if the price is $15$ dollars per ton. Assuming the supply function is of the form $y=mx+b$, find the slope, $m$ and y-intercept, $b$ $m$: [ANS]\n$b$: [ANS]", "answer_v2": [ "0.00404761904761905", "-235.952380952381" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose a mining company will supply $92000$ tons of ore per month if the price is $90$ dollars per ton but will supply $64000$ tons per month if the price is $20$ dollars per ton. Assuming the supply function is of the form $y=mx+b$, find the slope, $m$ and y-intercept, $b$ $m$: [ANS]\n$b$: [ANS]", "answer_v3": [ "0.0025", "-140" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0298", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "algebra" ], "problem_v1": "You are working on a new temperature scale that will unify the earth. After some thought you decide to call it the-universal-scale. Let $F$ denote the temperature in degrees Fahrenheit, and let $X$ denote your new temperature scale. You want it to be such that if $F=0$ then $X=18$ and if $F=100$ then $X=134$. You also want $X$ to be such that if you plot $X$ against $F$ you obtain a straight line. You obtain the formula $X=m F+b$ where $m$=[ANS]\nand $b$=[ANS]\nHint: Think of this problem as one of finding the slope-intercept form of a straight line in the $(F,X)$ plane given two points on the line.", "answer_v1": [ "(134-18)/100", "18" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "You are working on a new temperature scale that will unify the earth. After some thought you decide to call it the-universal-scale. Let $F$ denote the temperature in degrees Fahrenheit, and let $X$ denote your new temperature scale. You want it to be such that if $F=0$ then $X=8$ and if $F=100$ then $X=118$. You also want $X$ to be such that if you plot $X$ against $F$ you obtain a straight line. You obtain the formula $X=m F+b$ where $m$=[ANS]\nand $b$=[ANS]\nHint: Think of this problem as one of finding the slope-intercept form of a straight line in the $(F,X)$ plane given two points on the line.", "answer_v2": [ "(118-8)/100", "8" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "You are working on a new temperature scale that will unify the earth. After some thought you decide to call it the-universal-scale. Let $F$ denote the temperature in degrees Fahrenheit, and let $X$ denote your new temperature scale. You want it to be such that if $F=0$ then $X=10$ and if $F=100$ then $X=127$. You also want $X$ to be such that if you plot $X$ against $F$ you obtain a straight line. You obtain the formula $X=m F+b$ where $m$=[ANS]\nand $b$=[ANS]\nHint: Think of this problem as one of finding the slope-intercept form of a straight line in the $(F,X)$ plane given two points on the line.", "answer_v3": [ "(127-10)/100", "10" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0299", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "algebra" ], "problem_v1": "Suppose the next time you buy a house including the land it sits on for 165000 Dollars. The real estate agent tells you that the land costs 17000 more than the house. You apply your formula derived in the previous problem. To do so you set $S=$ [ANS] dollars and $D=$ [ANS] dollars. and you find out that the price of the house is [ANS] dollars and the price of the land is [ANS] dollars.", "answer_v1": [ "165000", "17000", "74000", "91000" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Suppose the next time you buy a house including the land it sits on for 125000 Dollars. The real estate agent tells you that the land costs 24000 more than the house. You apply your formula derived in the previous problem. To do so you set $S=$ [ANS] dollars and $D=$ [ANS] dollars. and you find out that the price of the house is [ANS] dollars and the price of the land is [ANS] dollars.", "answer_v2": [ "125000", "24000", "50500", "74500" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Suppose the next time you buy a house including the land it sits on for 139000 Dollars. The real estate agent tells you that the land costs 17000 more than the house. You apply your formula derived in the previous problem. To do so you set $S=$ [ANS] dollars and $D=$ [ANS] dollars. and you find out that the price of the house is [ANS] dollars and the price of the land is [ANS] dollars.", "answer_v3": [ "139000", "17000", "61000", "78000" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Algebra_0300", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "algebra" ], "problem_v1": "After a 5\\% raise your new salary is \\$ 84000. Before the raise your salary was \\$ [ANS].", "answer_v1": [ "80000" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "After a 5\\% raise your new salary is \\$ 26250. Before the raise your salary was \\$ [ANS].", "answer_v2": [ "25000" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "After a 5\\% raise your new salary is \\$ 47250. Before the raise your salary was \\$ [ANS].", "answer_v3": [ "45000" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0301", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "algebra" ], "problem_v1": "A new apartment building was sold for \\$165000 7 years after it was purchased. The original owners calculated that the building appreciated \\$7000 per year while they owned it. Find a linear function that describes the appreciation of the building, if $x$ is a number of years since the original purchase. $f(x)=$ [ANS]", "answer_v1": [ "7000(x-7)+165000" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "A new apartment building was sold for \\$95000 3 years after it was purchased. The original owners calculated that the building appreciated \\$10000 per year while they owned it. Find a linear function that describes the appreciation of the building, if $x$ is a number of years since the original purchase. $f(x)=$ [ANS]", "answer_v2": [ "10000(x-3)+95000" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "A new apartment building was sold for \\$120000 4 years after it was purchased. The original owners calculated that the building appreciated \\$7000 per year while they owned it. Find a linear function that describes the appreciation of the building, if $x$ is a number of years since the original purchase. $f(x)=$ [ANS]", "answer_v3": [ "7000(x-4)+120000" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0302", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "algebra" ], "problem_v1": "The business opened with a debt of \\$4300. After 4 years, it accumulated profit of \\$3800. Find the profit as a function of time $t$, knowing the profit function is linear. $P(t)=$ [ANS]", "answer_v1": [ "(3800+4300)/4*t-4300" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "The business opened with a debt of \\$2200. After 2 years, it accumulated profit of \\$4800. Find the profit as a function of time $t$, knowing the profit function is linear. $P(t)=$ [ANS]", "answer_v2": [ "(4800+2200)/2*t-2200" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "The business opened with a debt of \\$2900. After 3 years, it accumulated profit of \\$3800. Find the profit as a function of time $t$, knowing the profit function is linear. $P(t)=$ [ANS]", "answer_v3": [ "(3800+2900)/3*t-2900" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0303", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "algebra" ], "problem_v1": "You work in a lab. One day you need 616 oz of a chemical solution consisting of three parts alcohol and five parts acid. How much of each should be used? Answer: [ANS] oz of alcohol and [ANS] oz of acid", "answer_v1": [ "231", "385" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "You work in a lab. One day you need 144 oz of a chemical solution consisting of three parts alcohol and five parts acid. How much of each should be used? Answer: [ANS] oz of alcohol and [ANS] oz of acid", "answer_v2": [ "54", "90" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "You work in a lab. One day you need 304 oz of a chemical solution consisting of three parts alcohol and five parts acid. How much of each should be used? Answer: [ANS] oz of alcohol and [ANS] oz of acid", "answer_v3": [ "114", "190" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0304", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "2", "keywords": [ "algebra" ], "problem_v1": "A company produce snowboards. Fixed costs are \\$1320 and the cost per snowboard is \\$260. An order has been placed for 6 snowboards. What should the retail price be in order for the company to break even? Answer: \\$ [ANS]", "answer_v1": [ "480" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A company produce snowboards. Fixed costs are \\$300 and the cost per snowboard is \\$300. An order has been placed for 3 snowboards. What should the retail price be in order for the company to break even? Answer: \\$ [ANS]", "answer_v2": [ "400" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A company produce snowboards. Fixed costs are \\$680 and the cost per snowboard is \\$260. An order has been placed for 4 snowboards. What should the retail price be in order for the company to break even? Answer: \\$ [ANS]", "answer_v3": [ "430" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0305", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "algebra" ], "problem_v1": "63\\% or 504 employees in a company are female. How many are male? Answer: [ANS]", "answer_v1": [ "296" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "29\\% or 290 employees in a company are female. How many are male? Answer: [ANS]", "answer_v2": [ "710" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "40\\% or 320 employees in a company are female. How many are male? Answer: [ANS]", "answer_v3": [ "480" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0306", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "algebra" ], "problem_v1": "A manufacturer has 3400 units in stock. The product is now selling at \\$6 per unit. Next month the unit price will increase by \\$0.50. The manufacturer wants the total revenue received from the sale of the 3400 units to be no less than \\$21200. What is the number of units that can be sold this month? Answer: At most [ANS] units", "answer_v1": [ "1800" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A manufacturer has 3000 units in stock. The product is now selling at \\$3 per unit. Next month the unit price will increase by \\$0.50. The manufacturer wants the total revenue received from the sale of the 3000 units to be no less than \\$10000. What is the number of units that can be sold this month? Answer: At most [ANS] units", "answer_v2": [ "1000" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A manufacturer has 2900 units in stock. The product is now selling at \\$4 per unit. Next month the unit price will increase by \\$0.50. The manufacturer wants the total revenue received from the sale of the 2900 units to be no less than \\$12400. What is the number of units that can be sold this month? Answer: At most [ANS] units", "answer_v3": [ "1300" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0307", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "algebra", "lines" ], "problem_v1": "A company manufactures and sells bookcases. The selling price is $57.9$ (in dollars) per bookcase. The total cost function is linear, and costs amount to $50,000$ (in dollars) for $2000$ bookcases and $32,120$ (in dollars) for $800$ bookcases. Find the break-even point. $($ [ANS], [ANS] $)$", "answer_v1": [ "469.767441860465", "27199.5348837209" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A company manufactures and sells bookcases. The selling price is $50.9$ (in dollars) per bookcase. The total cost function is linear, and costs amount to $50,000$ (in dollars) for $2000$ bookcases and $32,120$ (in dollars) for $800$ bookcases. Find the break-even point. $($ [ANS], [ANS] $)$", "answer_v2": [ "561.111111111111", "28560.5555555556" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A company manufactures and sells bookcases. The selling price is $53.9$ (in dollars) per bookcase. The total cost function is linear, and costs amount to $50,000$ (in dollars) for $2000$ bookcases and $32,120$ (in dollars) for $800$ bookcases. Find the break-even point. $($ [ANS], [ANS] $)$", "answer_v3": [ "517.948717948718", "27917.4358974359" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0308", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "linear modeling" ], "problem_v1": "A barnyard is full of chickens and pigs, and the total number of chicken feet and pig feet is $70$.\n(a) If there are $c$ chickens and $p$ pigs in the barnyard, write an equation relating the number of chickens and pigs to the total number of feet in the barnyard. [ANS]\n(b) If there are $13$ chickens, how many pigs are there?\nNumber of pigs=[ANS]", "answer_v1": [ "2*c+4*p = 70", "11" ], "answer_type_v1": [ "EQ", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A barnyard is full of chickens and pigs, and the total number of chicken feet and pig feet is $70$.\n(a) If there are $c$ chickens and $p$ pigs in the barnyard, write an equation relating the number of chickens and pigs to the total number of feet in the barnyard. [ANS]\n(b) If there are $5$ chickens, how many pigs are there?\nNumber of pigs=[ANS]", "answer_v2": [ "2*c+4*p = 70", "15" ], "answer_type_v2": [ "EQ", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A barnyard is full of chickens and pigs, and the total number of chicken feet and pig feet is $60$.\n(a) If there are $c$ chickens and $p$ pigs in the barnyard, write an equation relating the number of chickens and pigs to the total number of feet in the barnyard. [ANS]\n(b) If there are $8$ chickens, how many pigs are there?\nNumber of pigs=[ANS]", "answer_v3": [ "2*c+4*p = 60", "11" ], "answer_type_v3": [ "EQ", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0309", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "linear modeling" ], "problem_v1": "The coffee variety Arabica yields about 850 kg of coffee beans per hectare, while Robusta yields about 1200 kg per hectare. Suppose that a plantation has $a$ hectares of Arabica and $r$ hectares of Robusta. Robusta. Write an equation relating $a$ and $r$ if the plantation yields 35000 kg of coffee. Do not enter any commas or units in your answer. [ANS]", "answer_v1": [ "850*a+1200*r = 35000" ], "answer_type_v1": [ "EQ" ], "options_v1": [ [] ], "problem_v2": "The coffee variety Arabica yields about 700 kg of coffee beans per hectare, while Robusta yields about 1300 kg per hectare. Suppose that a plantation has $a$ hectares of Arabica and $r$ hectares of Robusta. Robusta. Write an equation relating $a$ and $r$ if the plantation yields 15000 kg of coffee. Do not enter any commas or units in your answer. [ANS]", "answer_v2": [ "700*a+1300*r = 15000" ], "answer_type_v2": [ "EQ" ], "options_v2": [ [] ], "problem_v3": "The coffee variety Arabica yields about 750 kg of coffee beans per hectare, while Robusta yields about 1250 kg per hectare. Suppose that a plantation has $a$ hectares of Arabica and $r$ hectares of Robusta. Robusta. Write an equation relating $a$ and $r$ if the plantation yields 20000 kg of coffee. Do not enter any commas or units in your answer. [ANS]", "answer_v3": [ "750*a+1250*r = 20000" ], "answer_type_v3": [ "EQ" ], "options_v3": [ [] ] }, { "id": "Algebra_0310", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "linear equations" ], "problem_v1": "The tuition cost for part-time students taking $C$ credits at Stonewall College is given by $550+300 C$ dollars.\n(a) Find the tuition cost for 13 credits. Do not use dollar signs or commas in your answer. \\$ [ANS]\n(b) If the tuition cost is \\$4750, how many credits are taken? [ANS]", "answer_v1": [ "4450", "14" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The tuition cost for part-time students taking $C$ credits at Stonewall College is given by $300+400 C$ dollars.\n(a) Find the tuition cost for 9 credits. Do not use dollar signs or commas in your answer. \\$ [ANS]\n(b) If the tuition cost is \\$4700, how many credits are taken? [ANS]", "answer_v2": [ "3900", "11" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The tuition cost for part-time students taking $C$ credits at Stonewall College is given by $400+350 C$ dollars.\n(a) Find the tuition cost for 10 credits. Do not use dollar signs or commas in your answer. \\$ [ANS]\n(b) If the tuition cost is \\$4600, how many credits are taken? [ANS]", "answer_v3": [ "3900", "12" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0311", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "Equations" ], "problem_v1": "(a) Eric plans to spend \\$12.25 on ice cream cones at \\$1.75 each. Write an equation whose solution is the number of ice cream cones he can buy. Use $n$ for the number of ice cream cones. You may asssume there is no tax. [ANS]\n(b) Solve your equation above for the number of ice cream cones $n$. $n=$ [ANS]", "answer_v1": [ "12.25 = 1.75*n", "7" ], "answer_type_v1": [ "EQ", "NV" ], "options_v1": [ [], [] ], "problem_v2": "(a) Eric plans to spend \\$10.00 on ice cream cones at \\$1.25 each. Write an equation whose solution is the number of ice cream cones he can buy. Use $n$ for the number of ice cream cones. You may asssume there is no tax. [ANS]\n(b) Solve your equation above for the number of ice cream cones $n$. $n=$ [ANS]", "answer_v2": [ "10 = 1.25*n", "8" ], "answer_type_v2": [ "EQ", "NV" ], "options_v2": [ [], [] ], "problem_v3": "(a) Eric plans to spend \\$8.75 on ice cream cones at \\$1.25 each. Write an equation whose solution is the number of ice cream cones he can buy. Use $n$ for the number of ice cream cones. You may asssume there is no tax. [ANS]\n(b) Solve your equation above for the number of ice cream cones $n$. $n=$ [ANS]", "answer_v3": [ "8.75 = 1.25*n", "7" ], "answer_type_v3": [ "EQ", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0312", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "linear functions" ], "problem_v1": "The temperature of the soil is $28 \\ {}^{\\circ}\\mathrm{C}$ at the surface and decreases by $0.07 \\ {}^{\\circ}\\mathrm{C}$ for each centimeter below the surface. Express temperature $T$, in degrees Celsius, as a function of depth $d$, in centimeters, below the surface. Enter your answer as an equation with $T$ on the left side, and an expression involving $d$ on the right. [ANS]", "answer_v1": [ "T = 28-0.07*d" ], "answer_type_v1": [ "EQ" ], "options_v1": [ [] ], "problem_v2": "The temperature of the soil is $30 \\ {}^{\\circ}\\mathrm{C}$ at the surface and decreases by $0.01 \\ {}^{\\circ}\\mathrm{C}$ for each centimeter below the surface. Express temperature $T$, in degrees Celsius, as a function of depth $d$, in centimeters, below the surface. Enter your answer as an equation with $T$ on the left side, and an expression involving $d$ on the right. [ANS]", "answer_v2": [ "T = 30-0.01*d" ], "answer_type_v2": [ "EQ" ], "options_v2": [ [] ], "problem_v3": "The temperature of the soil is $28 \\ {}^{\\circ}\\mathrm{C}$ at the surface and decreases by $0.03 \\ {}^{\\circ}\\mathrm{C}$ for each centimeter below the surface. Express temperature $T$, in degrees Celsius, as a function of depth $d$, in centimeters, below the surface. Enter your answer as an equation with $T$ on the left side, and an expression involving $d$ on the right. [ANS]", "answer_v3": [ "T = 28-0.03*d" ], "answer_type_v3": [ "EQ" ], "options_v3": [ [] ] }, { "id": "Algebra_0313", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "linear functions" ], "problem_v1": "You buy a saguaro cactus 7 feet high and it grows at a rate of 0.8 inches each year. Express its height $h$, in inches, as a function of time $t$, in years, since the purchase. Enter your answer as an equation with $h$ on the left side, and an expression involving $t$ on the right. [ANS]", "answer_v1": [ "h = 84+0.8*t" ], "answer_type_v1": [ "EQ" ], "options_v1": [ [] ], "problem_v2": "You buy a saguaro cactus 9 feet high and it grows at a rate of 0.2 inches each year. Express its height $h$, in inches, as a function of time $t$, in years, since the purchase. Enter your answer as an equation with $h$ on the left side, and an expression involving $t$ on the right. [ANS]", "answer_v2": [ "h = 108+0.2*t" ], "answer_type_v2": [ "EQ" ], "options_v2": [ [] ], "problem_v3": "You buy a saguaro cactus 8 feet high and it grows at a rate of 0.4 inches each year. Express its height $h$, in inches, as a function of time $t$, in years, since the purchase. Enter your answer as an equation with $h$ on the left side, and an expression involving $t$ on the right. [ANS]", "answer_v3": [ "h = 96+0.4*t" ], "answer_type_v3": [ "EQ" ], "options_v3": [ [] ] }, { "id": "Algebra_0314", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "linear functions" ], "problem_v1": "A homing pigeon starts 800 miles from home and flies 50 miles toward home each day. Express the distance from home $D$, in miles, as a function of the number of days $d$. Enter your answer as an equation with $D$ on the left side, and an expression involving $d$ on the right. [ANS]", "answer_v1": [ "D = 800-50*d" ], "answer_type_v1": [ "EQ" ], "options_v1": [ [] ], "problem_v2": "A homing pigeon starts 500 miles from home and flies 25 miles toward home each day. Express the distance from home $D$, in miles, as a function of the number of days $d$. Enter your answer as an equation with $D$ on the left side, and an expression involving $d$ on the right. [ANS]", "answer_v2": [ "D = 500-25*d" ], "answer_type_v2": [ "EQ" ], "options_v2": [ [] ], "problem_v3": "A homing pigeon starts 400 miles from home and flies 25 miles toward home each day. Express the distance from home $D$, in miles, as a function of the number of days $d$. Enter your answer as an equation with $D$ on the left side, and an expression involving $d$ on the right. [ANS]", "answer_v3": [ "D = 400-25*d" ], "answer_type_v3": [ "EQ" ], "options_v3": [ [] ] }, { "id": "Algebra_0315", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "algebra", "inequality" ], "problem_v1": "As dry air moves upward, it expands and in so doing cools at a rate of about $1^{0} C$ for 100m rise, up to about 12 km. a) If the ground temperature is $23 ^{0}$, write a formula for the temperature at height x km.\nAnswer: [ANS]\nb) What range of temperature can be expected if a plane takes off and reaches a maximum height of 8 km.Write answer in interval notation.\nAnswer: [ANS]", "answer_v1": [ "23-10*x", "[-57,23]" ], "answer_type_v1": [ "EX", "INT" ], "options_v1": [ [], [] ], "problem_v2": "As dry air moves upward, it expands and in so doing cools at a rate of about $1^{0} C$ for 100m rise, up to about 12 km. a) If the ground temperature is $15 ^{0}$, write a formula for the temperature at height x km.\nAnswer: [ANS]\nb) What range of temperature can be expected if a plane takes off and reaches a maximum height of 10 km.Write answer in interval notation.\nAnswer: [ANS]", "answer_v2": [ "15-10*x", "[-85,15]" ], "answer_type_v2": [ "EX", "INT" ], "options_v2": [ [], [] ], "problem_v3": "As dry air moves upward, it expands and in so doing cools at a rate of about $1^{0} C$ for 100m rise, up to about 12 km. a) If the ground temperature is $18 ^{0}$, write a formula for the temperature at height x km.\nAnswer: [ANS]\nb) What range of temperature can be expected if a plane takes off and reaches a maximum height of 8 km.Write answer in interval notation.\nAnswer: [ANS]", "answer_v3": [ "18-10*x", "[-62,18]" ], "answer_type_v3": [ "EX", "INT" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0316", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "equation", "line" ], "problem_v1": "Initially, you have lost 20,000 dollars in the stock market and you continue to lose 350 dollars per month. In how many months will it be before your losses total 36300 dollars, thus your balance is-36300? Number of months is=[ANS]", "answer_v1": [ "46.5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Initially, you have lost 20,000 dollars in the stock market and you continue to lose 350 dollars per month. In how many months will it be before your losses total 26240 dollars, thus your balance is-26240? Number of months is=[ANS]", "answer_v2": [ "17.5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Initially, you have lost 20,000 dollars in the stock market and you continue to lose 350 dollars per month. In how many months will it be before your losses total 29700 dollars, thus your balance is-29700? Number of months is=[ANS]", "answer_v3": [ "27.5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0317", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "2", "keywords": [ "Statistics", "Linear regression", "line" ], "problem_v1": "The table below contains the average public school classroom teacher's salaries, $S$, for an 11-year period. Letting $t=0$ represent 1990, use a graphing utility to find a linear model for the data.\n$\\begin{array}{ccccccc}\\hline Year & 1990 & 1991 & 1992 & 1993 & 1994 & 1995 \\\\ \\hline Salary & 33592 & 34887 & 34370 & 35394 & 37188 & 38158 \\\\ \\hline \\end{array}$\n$\\begin{array}{cccccc}\\hline Year & 1996 & 1997 & 1998 & 1999 & 2000 \\\\ \\hline Salary & 38313 & 39728 & 41309 & 40667 & 42593 \\\\ \\hline \\end{array}$\nSalary, written as a function of $t$ is given by\n$S(t)=$ [ANS]", "answer_v1": [ "33348.3181818182+897.590909090909*t" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "The table below contains the average public school classroom teacher's salaries, $S$, for an 11-year period. Letting $t=0$ represent 1990, use a graphing utility to find a linear model for the data.\n$\\begin{array}{ccccccc}\\hline Year & 1990 & 1991 & 1992 & 1993 & 1994 & 1995 \\\\ \\hline Salary & 32041 & 33563 & 36618 & 35316 & 35730 & 37174 \\\\ \\hline \\end{array}$\n$\\begin{array}{cccccc}\\hline Year & 1996 & 1997 & 1998 & 1999 & 2000 \\\\ \\hline Salary & 38996 & 37959 & 41166 & 41274 & 43761 \\\\ \\hline \\end{array}$\nSalary, written as a function of $t$ is given by\n$S(t)=$ [ANS]", "answer_v2": [ "32525.2727272727+1014.90909090909*t" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "The table below contains the average public school classroom teacher's salaries, $S$, for an 11-year period. Letting $t=0$ represent 1990, use a graphing utility to find a linear model for the data.\n$\\begin{array}{ccccccc}\\hline Year & 1990 & 1991 & 1992 & 1993 & 1994 & 1995 \\\\ \\hline Salary & 32457 & 34286 & 34019 & 35423 & 38044 & 39389 \\\\ \\hline \\end{array}$\n$\\begin{array}{cccccc}\\hline Year & 1996 & 1997 & 1998 & 1999 & 2000 \\\\ \\hline Salary & 40207 & 38501 & 39765 & 40513 & 40534 \\\\ \\hline \\end{array}$\nSalary, written as a function of $t$ is given by\n$S(t)=$ [ANS]", "answer_v3": [ "33428.4545454545+825.909090909091*t" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0318", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "equation", "line" ], "problem_v1": "Teaneck High had an enrollment of 11510 in 1986 and an enrollment of 12910 in year 1998. What is the predicted enrollment of Teaneck High in 2007 if we assume a linear model? Predicted Enrollment=[ANS]", "answer_v1": [ "13960" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Teaneck High had an enrollment of 10160 in 1981 and an enrollment of 14660 in year 1994. What is the predicted enrollment of Teaneck High in 2019 if we assume a linear model? Predicted Enrollment=[ANS]", "answer_v2": [ "23313.8461538462" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Teaneck High had an enrollment of 10630 in 1983 and an enrollment of 13030 in year 1997. What is the predicted enrollment of Teaneck High in 2005 if we assume a linear model? Predicted Enrollment=[ANS]", "answer_v3": [ "14401.4285714286" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0319", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "equation", "line" ], "problem_v1": "A store is offering a 15\\% discount on all items. Write an equation relating the sale price $S$ for an item to its list price $L$. [ANS]", "answer_v1": [ "S=L-15/100 * L" ], "answer_type_v1": [ "EQ" ], "options_v1": [ [] ], "problem_v2": "A store is offering a 3\\% discount on all items. Write an equation relating the sale price $S$ for an item to its list price $L$. [ANS]", "answer_v2": [ "S=L-3/100 * L" ], "answer_type_v2": [ "EQ" ], "options_v2": [ [] ], "problem_v3": "A store is offering a 7\\% discount on all items. Write an equation relating the sale price $S$ for an item to its list price $L$. [ANS]", "answer_v3": [ "S=L-7/100 * L" ], "answer_type_v3": [ "EQ" ], "options_v3": [ [] ] }, { "id": "Algebra_0320", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "equation", "line" ], "problem_v1": "A manufacturer pays its assembly line workers \\$11.51 per hour. In addition, workers receive a piece work rate of \\$0.83 per unit produced. Write a linear equation for the hourly wages $W$ in terms of the number of units $x$ produced per hour. linear equation: $W=$ [ANS]\nWhat is the hourly wage for Mike, who produces 23 units in one hour? Mike's wage=[ANS]", "answer_v1": [ "0.83*x + 11.51", "30.6" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A manufacturer pays its assembly line workers \\$10.16 per hour. In addition, workers receive a piece work rate of \\$1.19 per unit produced. Write a linear equation for the hourly wages $W$ in terms of the number of units $x$ produced per hour. linear equation: $W=$ [ANS]\nWhat is the hourly wage for Mike, who produces 13 units in one hour? Mike's wage=[ANS]", "answer_v2": [ "1.19*x + 10.16", "25.63" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A manufacturer pays its assembly line workers \\$10.63 per hour. In addition, workers receive a piece work rate of \\$0.86 per unit produced. Write a linear equation for the hourly wages $W$ in terms of the number of units $x$ produced per hour. linear equation: $W=$ [ANS]\nWhat is the hourly wage for Mike, who produces 15 units in one hour? Mike's wage=[ANS]", "answer_v3": [ "0.86*x + 10.63", "23.53" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0321", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "word problem", "revenue" ], "problem_v1": "A gas station sells 1400 gallons of gasoline per hour if it charges \\$ 2.20 per gallon but only 1300 gallons per hour if it charges \\$ 2.45 per gallon. Assuming a linear model\n(a) How many gallons would be sold per hour of the price is \\$ 2.35 per gallon? Answer: [ANS]\n(b) What must the gasoline price be in order to sell 1400 gallons per hour? Answer: \\$ [ANS]\n(c) Compute the revenue taken at the four prices mentioned in this problem--\\$ 2.20, \\$ 2.35, \\$ 2.45 and your answer to part (b). Which price gives the most revenue? Answer: \\$ [ANS]\n(d) What is the price that the gas station should charge to maximize revenue? Answer: \\$ [ANS]", "answer_v1": [ "1340", "2.2", "2.45", "2.85" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "A gas station sells 1600 gallons of gasoline per hour if it charges \\$ 2.45 per gallon but only 1000 gallons per hour if it charges \\$ 2.90 per gallon. Assuming a linear model\n(a) How many gallons would be sold per hour of the price is \\$ 2.55 per gallon? Answer: [ANS]\n(b) What must the gasoline price be in order to sell 1600 gallons per hour? Answer: \\$ [ANS]\n(c) Compute the revenue taken at the four prices mentioned in this problem--\\$ 2.45, \\$ 2.55, \\$ 2.90 and your answer to part (b). Which price gives the most revenue? Answer: \\$ [ANS]\n(d) What is the price that the gas station should charge to maximize revenue? Answer: \\$ [ANS]", "answer_v2": [ "1466.66666666667", "2.45", "2.45", "1.825" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "A gas station sells 1500 gallons of gasoline per hour if it charges \\$ 2.15 per gallon but only 900 gallons per hour if it charges \\$ 2.60 per gallon. Assuming a linear model\n(a) How many gallons would be sold per hour of the price is \\$ 2.30 per gallon? Answer: [ANS]\n(b) What must the gasoline price be in order to sell 1300 gallons per hour? Answer: \\$ [ANS]\n(c) Compute the revenue taken at the four prices mentioned in this problem--\\$ 2.15, \\$ 2.30, \\$ 2.60 and your answer to part (b). Which price gives the most revenue? Answer: \\$ [ANS]\n(d) What is the price that the gas station should charge to maximize revenue? Answer: \\$ [ANS]", "answer_v3": [ "1300", "2.3", "2.15", "1.6375" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Algebra_0322", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "2", "keywords": [ "Lines", "Linear models" ], "problem_v1": "An online company sells handmade samurai katana swords. The website costs \\$ 390 a month to maintain. Each katana costs \\$ 215 to make, and they sell each katana for \\$ 685. A. Create a linear model in the form $y=m x+b$ where $x$ is the number of swords sold per month and $y$ is the net monthly profit. [ANS]\nB. Using this model, find the number of swords that would need to be sold per month to have a monthly profit of \\$ 4170. [ANS]", "answer_v1": [ "y=470*x-390", "9.70212765957447" ], "answer_type_v1": [ "EQ", "NV" ], "options_v1": [ [], [] ], "problem_v2": "An online company sells handmade samurai katana swords. The website costs \\$ 85 a month to maintain. Each katana costs \\$ 290 to make, and they sell each katana for \\$ 545. A. Create a linear model in the form $y=m x+b$ where $x$ is the number of swords sold per month and $y$ is the net monthly profit. [ANS]\nB. Using this model, find the number of swords that would need to be sold per month to have a monthly profit of \\$ 3000. [ANS]", "answer_v2": [ "y=255*x-85", "12.0980392156863" ], "answer_type_v2": [ "EQ", "NV" ], "options_v2": [ [], [] ], "problem_v3": "An online company sells handmade samurai katana swords. The website costs \\$ 190 a month to maintain. Each katana costs \\$ 220 to make, and they sell each katana for \\$ 580. A. Create a linear model in the form $y=m x+b$ where $x$ is the number of swords sold per month and $y$ is the net monthly profit. [ANS]\nB. Using this model, find the number of swords that would need to be sold per month to have a monthly profit of \\$ 3650. [ANS]", "answer_v3": [ "y=360*x-190", "10.6666666666667" ], "answer_type_v3": [ "EQ", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0323", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "cost", "revenue", "break-even", "business", "cost", "application" ], "problem_v1": "Bea T. Howen, a sophomore college student, lost her scholarship after receiving a D in her \"Music Appreciation\" course. She decided to buy a snow plow to supplement her income during the winter months. It cost her \\$6750.00. Fuel and standard maintenance will cost her an additional \\$9.00 for each hour of use. Find the cost function $C(x)$ associated with operating the snow plow for $x$ hours. $C(x)=$ [ANS]\nIf she charges \\$36.00 per hour write the revenue function $R(x)$ for the amount of revenue gained from $x$ hours of use. $R(x)=$ [ANS]\nFind the profit function $P(x)$ for the amount of profit gained from $x$ hours of use. $P(x)=$ [ANS]\nHow many hours will she need to work to break even? [ANS] hours", "answer_v1": [ "9*x+6750", "36*x", "36*x-9*x-6750", "250" ], "answer_type_v1": [ "EX", "EX", "EX", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Bea T. Howen, a sophomore college student, lost her scholarship after receiving a D in her \"Music Appreciation\" course. She decided to buy a snow plow to supplement her income during the winter months. It cost her \\$6400.00. Fuel and standard maintenance will cost her an additional \\$7.50 for each hour of use. Find the cost function $C(x)$ associated with operating the snow plow for $x$ hours. $C(x)=$ [ANS]\nIf she charges \\$39.50 per hour write the revenue function $R(x)$ for the amount of revenue gained from $x$ hours of use. $R(x)=$ [ANS]\nFind the profit function $P(x)$ for the amount of profit gained from $x$ hours of use. $P(x)=$ [ANS]\nHow many hours will she need to work to break even? [ANS] hours", "answer_v2": [ "7.5*x+6400", "39.5*x", "39.5*x-7.5*x-6400", "200" ], "answer_type_v2": [ "EX", "EX", "EX", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Bea T. Howen, a sophomore college student, lost her scholarship after receiving a D in her \"Music Appreciation\" course. She decided to buy a snow plow to supplement her income during the winter months. It cost her \\$5600.00. Fuel and standard maintenance will cost her an additional \\$8.00 for each hour of use. Find the cost function $C(x)$ associated with operating the snow plow for $x$ hours. $C(x)=$ [ANS]\nIf she charges \\$36.00 per hour write the revenue function $R(x)$ for the amount of revenue gained from $x$ hours of use. $R(x)=$ [ANS]\nFind the profit function $P(x)$ for the amount of profit gained from $x$ hours of use. $P(x)=$ [ANS]\nHow many hours will she need to work to break even? [ANS] hours", "answer_v3": [ "8*x+5600", "36*x", "36*x-8*x-5600", "200" ], "answer_type_v3": [ "EX", "EX", "EX", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Algebra_0324", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "Lines", "Modeling" ], "problem_v1": "It costs a beekeeper \\$2.45 to make each pint jar of honey, and she sells the jars to stores at a price of \\$3.5 each. Her other monthly costs are the lease of the land and building \\$840, general supplies and equipment \\$165, utilities \\$350, transportation \\$190 and fixed salaries \\$5000. What is the monthly break even point for the number of pints made and sold to the nearest integer? [ANS]\nThe beekeeper decides to give herself a raise amounting to \\$140 per month. What is the monthly break even point to the nearest integer after the change? [ANS]", "answer_v1": [ "6233", "6367" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "It costs a beekeeper \\$2.05 to make each pint jar of honey, and she sells the jars to stores at a price of \\$2.99 each. Her other monthly costs are the lease of the land and building \\$980, general supplies and equipment \\$115, utilities \\$290, transportation \\$300 and fixed salaries \\$4900. What is the monthly break even point for the number of pints made and sold to the nearest integer? [ANS]\nThe beekeeper decides to give herself a raise amounting to \\$70 per month. What is the monthly break even point to the nearest integer after the change? [ANS]", "answer_v2": [ "7005", "7080" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "It costs a beekeeper \\$2.19 to make each pint jar of honey, and she sells the jars to stores at a price of \\$3.4 each. Her other monthly costs are the lease of the land and building \\$840, general supplies and equipment \\$125, utilities \\$330, transportation \\$180 and fixed salaries \\$5000. What is the monthly break even point for the number of pints made and sold to the nearest integer? [ANS]\nThe beekeeper decides to give herself a raise amounting to \\$170 per month. What is the monthly break even point to the nearest integer after the change? [ANS]", "answer_v3": [ "5351", "5492" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0325", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "inequalities", "applications" ], "problem_v1": "Mike can be paid in one of two ways based on the amount of merchandise he sells: Plan A: A salary of $\\$1{,}100.00$ per month, plus a commission of $11\\%$ of sales, OR Plan B: A salary of $\\$1{,}450.00$ per month, plus a commission of $15\\%$ of sales in excess of $\\$8{,}000.00$. For what amount of monthly sales is plan B better than plan A if we can assume that Mike's sales are always more than $\\$8{,}000.00$? Write your answer an inequality involving $x$, where $x$ represents the total monthly sales. Answer: [ANS]", "answer_v1": [ "(-infinity, 21250)" ], "answer_type_v1": [ "INT" ], "options_v1": [ [] ], "problem_v2": "Mike can be paid in one of two ways based on the amount of merchandise he sells: Plan A: A salary of $\\$1{,}050.00$ per month, plus a commission of $10\\%$ of sales, OR Plan B: A salary of $\\$1{,}250.00$ per month, plus a commission of $15\\%$ of sales in excess of $\\$9{,}000.00$. For what amount of monthly sales is plan A better than plan B if we can assume that Mike's sales are always more than $\\$9{,}000.00$? Write your answer an inequality involving $x$, where $x$ represents the total monthly sales. Answer: [ANS]", "answer_v2": [ "(-infinity, 23000)" ], "answer_type_v2": [ "INT" ], "options_v2": [ [] ], "problem_v3": "Mike can be paid in one of two ways based on the amount of merchandise he sells: Plan A: A salary of $\\$950.00$ per month, plus a commission of $11\\%$ of sales, OR Plan B: A salary of $\\$1{,}250.00$ per month, plus a commission of $13\\%$ of sales in excess of $\\$6{,}000.00$. For what amount of monthly sales is plan A better than plan B if we can assume that Mike's sales are always more than $\\$6{,}000.00$? Write your answer an inequality involving $x$, where $x$ represents the total monthly sales. Answer: [ANS]", "answer_v3": [ "(-infinity, 24000)" ], "answer_type_v3": [ "INT" ], "options_v3": [ [] ] }, { "id": "Algebra_0326", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "problem solving", "linear", "equations" ], "problem_v1": "A survey consisting of $5500$ students on campus says that $50 \\%$ prefer the color blue over the color red. How many students prefer the color red? Answer: [ANS]", "answer_v1": [ "2750" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A survey consisting of $2800$ students on campus says that $75 \\%$ prefer the color red over the color blue. How many students prefer the color red? Answer: [ANS]", "answer_v2": [ "2100" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A survey consisting of $3700$ students on campus says that $50 \\%$ prefer the color red over the color blue. How many students prefer the color blue? Answer: [ANS]", "answer_v3": [ "1850" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0327", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "functions", "algebra", "application of linear equations" ], "problem_v1": "The average salary of a government worker is $\\$44{,}232.00$ per year. Given that this is about $14 \\%$ percent higher than the yearly salary of a private-sector worker, find the salary of the private-sector worker. Private-Sector Salary: [ANS] (Round your answer to the nearest cent and include units.)", "answer_v1": [ "38800.00" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The average salary of a government worker is $\\$42{,}126.00$ per year. Given that this is about $19 \\%$ percent higher than the yearly salary of a private-sector worker, find the salary of the private-sector worker. Private-Sector Salary: [ANS] (Round your answer to the nearest cent and include units.)", "answer_v2": [ "35400.00" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The average salary of a government worker is $\\$41{,}610.00$ per year. Given that this is about $14 \\%$ percent higher than the yearly salary of a private-sector worker, find the salary of the private-sector worker. Private-Sector Salary: [ANS] (Round your answer to the nearest cent and include units.)", "answer_v3": [ "36500.00" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0328", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "equations" ], "problem_v1": "A gallon of paint can cover 150 square feet. Find how many gallon containers of paint should be bought to paint 2 coats on each wall of a rectangular room with dimensions 16 feet by 10 feet and walls that are 10-feet tall. Answer: [ANS] gallons", "answer_v1": [ "7" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A gallon of paint can cover 100 square feet. Find how many gallon containers of paint should be bought to paint 3 coats on each wall of a rectangular room with dimensions 8 feet by 20 feet and walls that are 8-feet tall. Answer: [ANS] gallons", "answer_v2": [ "14" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A gallon of paint can cover 100 square feet. Find how many gallon containers of paint should be bought to paint 2 coats on each wall of a rectangular room with dimensions 20 feet by 10 feet and walls that are 8-feet tall. Answer: [ANS] gallons", "answer_v3": [ "10" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0329", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "functions", "algebra", "application of linear equations" ], "problem_v1": "In triangle $ABC$, angle $B$ is three times as large as angle $A$. The measure of angle $C$ is $20^{\\circ}$ more than that of angle $A$. Find the measures of the angles. $A$: [ANS] degrees $B$: [ANS] degrees $C$: [ANS] degrees", "answer_v1": [ "32", "96", "52" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "In triangle $ABC$, angle $B$ is twice as large as angle $A$. The measure of angle $C$ is $4^{\\circ}$ more than that of angle $A$. Find the measures of the angles. $A$: [ANS] degrees $B$: [ANS] degrees $C$: [ANS] degrees", "answer_v2": [ "44", "88", "48" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "In triangle $ABC$, angle $B$ is twice as large as angle $A$. The measure of angle $C$ is $20^{\\circ}$ more than that of angle $A$. Find the measures of the angles. $A$: [ANS] degrees $B$: [ANS] degrees $C$: [ANS] degrees", "answer_v3": [ "40", "80", "60" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0330", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "problem solving", "linear", "equations" ], "problem_v1": "The freshman class at a major university contains $5655$ students. This was a $13 \\%$ decrease from the size of the freshman class the year before. What was the size of last year's freshman class? Answer: [ANS]", "answer_v1": [ "6500" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The freshman class at a major university contains $4625$ students. This was a $25 \\%$ increase over the size of the freshman class the year before. What was the size of last year's freshman class? Answer: [ANS]", "answer_v2": [ "3700" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The freshman class at a major university contains $7020$ students. This was a $17 \\%$ increase over the size of the freshman class the year before. What was the size of last year's freshman class? Answer: [ANS]", "answer_v3": [ "6000" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0331", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "algebra", "linear equations" ], "problem_v1": "Solve $i=Prt$ for $t$, given that $P=\\$402$, $r=9$ \\%, and $i=\\$268$. Answer: $t=$ [ANS]", "answer_v1": [ "268/(0.01*402*9)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Solve $i=Prt$ for $t$, given that $P=\\$132$, $r=12$ \\%, and $i=\\$62$. Answer: $t=$ [ANS]", "answer_v2": [ "62/(0.01*132*12)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Solve $i=Prt$ for $t$, given that $P=\\$226$, $r=9$ \\%, and $i=\\$98$. Answer: $t=$ [ANS]", "answer_v3": [ "98/(0.01*226*9)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0332", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "algebra", "linear equations", "word problem", "modeling", "equations" ], "problem_v1": "A jeweler has five rings, each weighing $18$ grams, made of an alloy of $10$ \\% silver and $90$ \\% gold. He decides to melt down the rings and add enough silver to reduce the gold content to $70$ \\%. How much silver should he add? Ammount of Silver (in grams): [ANS]", "answer_v1": [ "(8100-90*70)/70" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A jeweler has five rings, each weighing $18$ grams, made of an alloy of $10$ \\% silver and $90$ \\% gold. He decides to melt down the rings and add enough silver to reduce the gold content to $40$ \\%. How much silver should he add? Ammount of Silver (in grams): [ANS]", "answer_v2": [ "(8100-90*40)/40" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A jeweler has five rings, each weighing $18$ grams, made of an alloy of $10$ \\% silver and $90$ \\% gold. He decides to melt down the rings and add enough silver to reduce the gold content to $50$ \\%. How much silver should he add? Ammount of Silver (in grams): [ANS]", "answer_v3": [ "(8100-90*50)/50" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0333", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "algebra", "linear equations", "word problem" ], "problem_v1": "A car leaves a town at $100$ kilometers per hour. How long will it take a second car, travelling at $125$ kilometers per hour, to catch the first car if it leaves $1$ hour later? Amount of Time (in hours): [ANS]", "answer_v1": [ "125/(125-100)-1" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A car leaves a town at $30$ kilometers per hour. How long will it take a second car, travelling at $135$ kilometers per hour, to catch the first car if it leaves $1$ hour later? Amount of Time (in hours): [ANS]", "answer_v2": [ "135/(135-30)-1" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A car leaves a town at $60$ kilometers per hour. How long will it take a second car, travelling at $105$ kilometers per hour, to catch the first car if it leaves $1$ hour later? Amount of Time (in hours): [ANS]", "answer_v3": [ "105/(105-60)-1" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0334", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "algebra", "linear equations", "mixing ratios" ], "problem_v1": "How many cups of grapefruit juice must be added to $50$ cups of a punch that is $5$ \\% grapefruit juice to obtain a punch that is $10$ \\% grapefruit juice? Amount (in cups): [ANS]", "answer_v1": [ "2.5/0.9" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "How many cups of grapefruit juice must be added to $20$ cups of a punch that is $8$ \\% grapefruit juice to obtain a punch that is $10$ \\% grapefruit juice? Amount (in cups): [ANS]", "answer_v2": [ "0.4/0.9" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "How many cups of grapefruit juice must be added to $30$ cups of a punch that is $6$ \\% grapefruit juice to obtain a punch that is $10$ \\% grapefruit juice? Amount (in cups): [ANS]", "answer_v3": [ "1.2/0.9" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0335", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "prealgebra", "common core", "formulas" ], "problem_v1": "The area inside a rectangle is its height times the width. A formula describing this relationship is $A=L w$. Your friend says his basketball goal comes with a rectangular concrete playing surface with area 651 square feet and width 31 feet. The length of the playing area for the game is [ANS] feet.", "answer_v1": [ "21" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The area inside a rectangle is its height times the width. A formula describing this relationship is $A=L w$. Your friend says his basketball goal comes with a rectangular concrete playing surface with area 630 square feet and width 35 feet. The length of the playing area for the game is [ANS] feet.", "answer_v2": [ "18" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The area inside a rectangle is its height times the width. A formula describing this relationship is $A=L w$. Your friend says his basketball goal comes with a rectangular concrete playing surface with area 589 square feet and width 31 feet. The length of the playing area for the game is [ANS] feet.", "answer_v3": [ "19" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0336", "subject": "Algebra", "topic": "Linear equations and functions", "subtopic": "Applications and models", "level": "2", "keywords": [ "prealgebra", "common core", "proportions" ], "problem_v1": "A microchip inspector found eight defective chips in a batch containing $700$ chips. At that rate, determine the number of defective chips in a batch of $25000$ chips. Write a proportion describing this problem in the form $\\frac{a}{b}=\\frac{x}{d}$ where $x$ represents the number that you are asked to find. [ANS]/[ANS]=[ANS]/[ANS] The number defective chips expected in the larger batch is $x=$ [ANS]", "answer_v1": [ "8", "700", "x", "25000", "285.714" ], "answer_type_v1": [ "NV", "NV", "EX", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "A microchip inspector found four defective chips in a batch containing $900$ chips. At that rate, determine the number of defective chips in a batch of $10000$ chips. Write a proportion describing this problem in the form $\\frac{a}{b}=\\frac{x}{d}$ where $x$ represents the number that you are asked to find. [ANS]/[ANS]=[ANS]/[ANS] The number defective chips expected in the larger batch is $x=$ [ANS]", "answer_v2": [ "4", "900", "x", "10000", "44.4444" ], "answer_type_v2": [ "NV", "NV", "EX", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "A microchip inspector found five defective chips in a batch containing $700$ chips. At that rate, determine the number of defective chips in a batch of $15000$ chips. Write a proportion describing this problem in the form $\\frac{a}{b}=\\frac{x}{d}$ where $x$ represents the number that you are asked to find. [ANS]/[ANS]=[ANS]/[ANS] The number defective chips expected in the larger batch is $x=$ [ANS]", "answer_v3": [ "5", "700", "x", "15000", "107.143" ], "answer_type_v3": [ "NV", "NV", "EX", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Algebra_0337", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Solve by factoring", "level": "3", "keywords": [ "polynomial equations" ], "problem_v1": "Use factoring to solve the polynomial equation: HINT: Try multiplying each side of the equation by a number to eliminate the denominators. \\frac{x^2}{240}+\\frac{x}{24}+\\frac{1}{10}=0 Answer: [ANS]", "answer_v1": [ "(-6, -4)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Use factoring to solve the polynomial equation: HINT: Try multiplying each side of the equation by a number to eliminate the denominators. \\frac{x^2}{20}+\\frac{x}{4}+\\frac{1}{5}=0 Answer: [ANS]", "answer_v2": [ "(-4, -1)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Use factoring to solve the polynomial equation: HINT: Try multiplying each side of the equation by a number to eliminate the denominators. \\frac{x^2}{48}+\\frac{x}{8}+\\frac{1}{6}=0 Answer: [ANS]", "answer_v3": [ "(-4, -2)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Algebra_0338", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Solve by factoring", "level": "2", "keywords": [ "algebra", "quadratic equation", "quadratic" ], "problem_v1": "Solve the following quadratic equation by factoring and applying the property: $ab=0$ if and only if $a=0$ or $b=0$. 8 n^2+11 n=0 Solutions (separate by commas): $n=$ [ANS]", "answer_v1": [ "(0, -1.375)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Solve the following quadratic equation by factoring and applying the property: $ab=0$ if and only if $a=0$ or $b=0$. 2 n^2+15 n=0 Solutions (separate by commas): $n=$ [ANS]", "answer_v2": [ "(0, -7.5)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Solve the following quadratic equation by factoring and applying the property: $ab=0$ if and only if $a=0$ or $b=0$. 2 n^2+13 n=0 Solutions (separate by commas): $n=$ [ANS]", "answer_v3": [ "(0, -6.5)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Algebra_0339", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Completing the square", "level": "2", "keywords": [ "calculus", "algebra", "absolute maximum/minimum", "completing the square", "maximum/minimum", "quadratic equations" ], "problem_v1": "Complete the square and find the minimum or maximum value of the quadratic function $y=5x^{2}+2x$.\n[ANS] 1. value is [ANS]", "answer_v1": [ "-1/5", "MINIMUM" ], "answer_type_v1": [ "NV", "MCS" ], "options_v1": [ [], [ "Minimum", "Maximum" ] ], "problem_v2": "Complete the square and find the minimum or maximum value of the quadratic function $y=9x-9x^{2}$.\n[ANS] 1. value is [ANS]", "answer_v2": [ "9/4", "MAXIMUM" ], "answer_type_v2": [ "NV", "MCS" ], "options_v2": [ [], [ "Minimum", "Maximum" ] ], "problem_v3": "Complete the square and find the minimum or maximum value of the quadratic function $y=2x-4x^{2}$.\n[ANS] 1. value is [ANS]", "answer_v3": [ "1/4", "MAXIMUM" ], "answer_type_v3": [ "NV", "MCS" ], "options_v3": [ [], [ "Minimum", "Maximum" ] ] }, { "id": "Algebra_0340", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Completing the square", "level": "1", "keywords": [ "logarithms" ], "problem_v1": "Solve $r^2-12 r+5=0$ by completing the square. If there is more than one correct answer, enter your answers as a comma separated list. If there are no answers, enter NONE.\n$r=$ [ANS]", "answer_v1": [ "(11.5678, 0.432236)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Solve $r^2-6 r+8=0$ by completing the square. If there is more than one correct answer, enter your answers as a comma separated list. If there are no answers, enter NONE.\n$r=$ [ANS]", "answer_v2": [ "(4, 2)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Solve $r^2-8 r+5=0$ by completing the square. If there is more than one correct answer, enter your answers as a comma separated list. If there are no answers, enter NONE.\n$r=$ [ANS]", "answer_v3": [ "(7.31662, 0.683375)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Algebra_0341", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Completing the square", "level": "1", "keywords": [ "completing the square" ], "problem_v1": "Complete the square by writing $-17+16x+x^2$ in the form $(x-h)^2+k$. Note: the numbers $h$ and $k$ can be positive or negative. $-17+16x+x^2=\\big($ [ANS] $\\big)^2+$ [ANS]", "answer_v1": [ "x+8", "-81" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Complete the square by writing $-19-10x+x^2$ in the form $(x-h)^2+k$. Note: the numbers $h$ and $k$ can be positive or negative. $-19-10x+x^2=\\big($ [ANS] $\\big)^2+$ [ANS]", "answer_v2": [ "x-5", "-44" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Complete the square by writing $-17-12x+x^2$ in the form $(x-h)^2+k$. Note: the numbers $h$ and $k$ can be positive or negative. $-17-12x+x^2=\\big($ [ANS] $\\big)^2+$ [ANS]", "answer_v3": [ "x-6", "-53" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0342", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Complex roots", "level": "2", "keywords": [ "algebra" ], "problem_v1": "Find all solutions in the complex numbers to the equation x^2-x-20=0 and enter them as a comma-separated list. If there is a repeated root, list it as many times as its multiplicity.\nSolutions: [ANS]", "answer_v1": [ "(5, -4)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Find all solutions in the complex numbers to the equation x^2+3x-10=0 and enter them as a comma-separated list. If there is a repeated root, list it as many times as its multiplicity.\nSolutions: [ANS]", "answer_v2": [ "(2, -5)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Find all solutions in the complex numbers to the equation x^2+x-12=0 and enter them as a comma-separated list. If there is a repeated root, list it as many times as its multiplicity.\nSolutions: [ANS]", "answer_v3": [ "(3, -4)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Algebra_0343", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Complex roots", "level": "2", "keywords": [ "complex", "imaginary", "quadratic", "root", "algebra", "complex number" ], "problem_v1": "Find all solutions of the equation $x^2+2x+7=0$ and express them in the form $a+b i$:\nsolutions: [ANS]\nNote: If there is more than one solution, enter a comma separated list (i.e.: 1+2i,3+4i). If there are no solutions, enter None.", "answer_v1": [ "(-1-2.44948974278318i, -1+2.44948974278318i)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Find all solutions of the equation $x^2-4x+8=0$ and express them in the form $a+b i$:\nsolutions: [ANS]\nNote: If there is more than one solution, enter a comma separated list (i.e.: 1+2i,3+4i). If there are no solutions, enter None.", "answer_v2": [ "(2-2i, 2+2i)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Find all solutions of the equation $x^2-2x+7=0$ and express them in the form $a+b i$:\nsolutions: [ANS]\nNote: If there is more than one solution, enter a comma separated list (i.e.: 1+2i,3+4i). If there are no solutions, enter None.", "answer_v3": [ "(1-2.44948974278318i, 1+2.44948974278318i)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Algebra_0344", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Solving equations", "level": "2", "keywords": [ "algebra", "quadratic equations" ], "problem_v1": "Find all real solutions of equation $2+7 z+z^2=0$.\nDoes the equation have real solutions? Input Yes or No: [ANS]\nIf your answer is Yes, input the solutions: $z_1=$ [ANS] and $z_2=$ [ANS] with $z_1\\le z_2$.", "answer_v1": [ "YES", "-6.70156211871642", "-0.298437881283576" ], "answer_type_v1": [ "TF", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find all real solutions of equation $3+4 z+z^2=0$.\nDoes the equation have real solutions? Input Yes or No: [ANS]\nIf your answer is Yes, input the solutions: $z_1=$ [ANS] and $z_2=$ [ANS] with $z_1\\le z_2$.", "answer_v2": [ "YES", "-3", "-1" ], "answer_type_v2": [ "TF", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find all real solutions of equation $2+5 z+z^2=0$.\nDoes the equation have real solutions? Input Yes or No: [ANS]\nIf your answer is Yes, input the solutions: $z_1=$ [ANS] and $z_2=$ [ANS] with $z_1\\le z_2$.", "answer_v3": [ "YES", "-4.56155281280883", "-0.43844718719117" ], "answer_type_v3": [ "TF", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0345", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Solving equations", "level": "3", "keywords": [ "algebra", "solve for variable", "solve for variable' 'fraction" ], "problem_v1": "Solve for $k$ from $0=151 (k-117x)^2-\\frac{x}{125}$, and give your answer as a comma-separated list.\n$k=$ [ANS].", "answer_v1": [ "(117*x-sqrt(x/18875), 117*x+sqrt(x/18875))" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Solve for $k$ from $0=18 (k-187x)^2-\\frac{x}{31}$, and give your answer as a comma-separated list.\n$k=$ [ANS].", "answer_v2": [ "(187*x-sqrt(x/558), 187*x+sqrt(x/558))" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Solve for $k$ from $0=64 (k-122x)^2-\\frac{x}{57}$, and give your answer as a comma-separated list.\n$k=$ [ANS].", "answer_v3": [ "(122*x-sqrt(x/3648), 122*x+sqrt(x/3648))" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Algebra_0346", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Solving equations", "level": "2", "keywords": [ "algebra", "solve for variable' 'fraction", "solve for variable" ], "problem_v1": "Write a quadratic function that has roots of 27 and P.\n$f(m)=$ [ANS]", "answer_v1": [ "(m - 27)*(m-P)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Write a quadratic function that has roots of-29 and P.\n$f(m)=$ [ANS]", "answer_v2": [ "(m - -29)*(m-P)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Write a quadratic function that has roots of-9 and P.\n$f(m)=$ [ANS]", "answer_v3": [ "(m - -9)*(m-P)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0347", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Solving equations", "level": "3", "keywords": [ "Algebra", "Functions", "Polynomial" ], "problem_v1": "Enter a quadratic polynomial which has roots at 5/6 and 3. [ANS]", "answer_v1": [ "a*(x-5/6)*(x-3)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Enter a quadratic polynomial which has roots at-17/19 and-7. [ANS]", "answer_v2": [ "a*(x--17/19)*(x--7)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Enter a quadratic polynomial which has roots at-8/13 and-5. [ANS]", "answer_v3": [ "a*(x--8/13)*(x--5)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0348", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Solving equations", "level": "3", "keywords": [ "power equations" ], "problem_v1": "Solve the equation $(x+5)^2+5=69$. If there is more than one correct answer, enter your answers as a comma separated list.\n$x$=[ANS]", "answer_v1": [ "(3, -13)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Solve the equation $(x+3)^2+6=31$. If there is more than one correct answer, enter your answers as a comma separated list.\n$x$=[ANS]", "answer_v2": [ "(2, -8)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Solve the equation $(x+3)^2+5=41$. If there is more than one correct answer, enter your answers as a comma separated list.\n$x$=[ANS]", "answer_v3": [ "(3, -9)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Algebra_0349", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Solving equations", "level": "2", "keywords": [ "algebra", "factoring", "quadratic", "difference of squares" ], "problem_v1": "Find all real number solutions for the equation 16y^{2}=49. Solutions (separate by commas): $y=$ [ANS]", "answer_v1": [ "(1.75, -1.75)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Find all real number solutions for the equation 4y^{2}=81. Solutions (separate by commas): $y=$ [ANS]", "answer_v2": [ "(4.5, -4.5)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Find all real number solutions for the equation 4y^{2}=64. Solutions (separate by commas): $y=$ [ANS]", "answer_v3": [ "(4, -4)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Algebra_0350", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Forms: vertex, factored, general", "level": "2", "keywords": [ "functions", "concavity", "quadratic", "zeros" ], "problem_v1": "Decide whether the function $f(t)=8 (t-1)(t+3)$ is a quadratic function. If so write the function in standard form, $f(t)=at^2+bt+c$. Below enter the proper values for the constants $a$, $b$, and $c$ once the function is put in standard form. If one of the constants is zero, be sure to enter 0 and do not leave any of answers blank. Account for subtraction by entering a negative value for the constant. If the function is not quadratic, enter, NA in each blank. $f(t)=$ [ANS] $t^2$+[ANS] $t$+[ANS]", "answer_v1": [ "8", "16", "-24" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Decide whether the function $f(t)=-14 (t-1)(t+3)$ is a quadratic function. If so write the function in standard form, $f(t)=at^2+bt+c$. Below enter the proper values for the constants $a$, $b$, and $c$ once the function is put in standard form. If one of the constants is zero, be sure to enter 0 and do not leave any of answers blank. Account for subtraction by entering a negative value for the constant. If the function is not quadratic, enter, NA in each blank. $f(t)=$ [ANS] $t^2$+[ANS] $t$+[ANS]", "answer_v2": [ "-14", "-28", "42" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Decide whether the function $f(t)=-6 (t+1)(t-3)$ is a quadratic function. If so write the function in standard form, $f(t)=at^2+bt+c$. Below enter the proper values for the constants $a$, $b$, and $c$ once the function is put in standard form. If one of the constants is zero, be sure to enter 0 and do not leave any of answers blank. Account for subtraction by entering a negative value for the constant. If the function is not quadratic, enter, NA in each blank. $f(t)=$ [ANS] $t^2$+[ANS] $t$+[ANS]", "answer_v3": [ "-6", "12", "18" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0351", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Forms: vertex, factored, general", "level": "2", "keywords": [ "functions", "concavity", "quadratic", "zeros" ], "problem_v1": "Simplify the function L(P)=(P+5)(5-P) by expanding and combining like terms. Simplify your answer as much as possible. $L(P)=$ [ANS]\nIs $L(P)$ a constant, linear, quadratic, or cubic function, or none of these? [ANS]", "answer_v1": [ "-P^2+25", "QUADRATIC" ], "answer_type_v1": [ "EX", "MCS" ], "options_v1": [ [], [ "constant", "linear", "quadratic", "cubic", "none of the above" ] ], "problem_v2": "Simplify the function L(P)=(P+2)(2-P) by expanding and combining like terms. Simplify your answer as much as possible. $L(P)=$ [ANS]\nIs $L(P)$ a constant, linear, quadratic, or cubic function, or none of these? [ANS]", "answer_v2": [ "-P^2+4", "QUADRATIC" ], "answer_type_v2": [ "EX", "MCS" ], "options_v2": [ [], [ "constant", "linear", "quadratic", "cubic", "none of the above" ] ], "problem_v3": "Simplify the function L(P)=(P+3)(3-P) by expanding and combining like terms. Simplify your answer as much as possible. $L(P)=$ [ANS]\nIs $L(P)$ a constant, linear, quadratic, or cubic function, or none of these? [ANS]", "answer_v3": [ "-P^2+9", "QUADRATIC" ], "answer_type_v3": [ "EX", "MCS" ], "options_v3": [ [], [ "constant", "linear", "quadratic", "cubic", "none of the above" ] ] }, { "id": "Algebra_0352", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Forms: vertex, factored, general", "level": "2", "keywords": [ "algebra", "solve for variable", "solve for variable' 'fraction" ], "problem_v1": "A quadratic function has its vertex at the point $(5,2)$. The function passes through the point $(3,5)$. When written in vertex form, the function is $f(x)=a(x-h)^2+k$, where:\n$a=$ [ANS].\n$h=$ [ANS].\n$k=$ [ANS].", "answer_v1": [ "0.75", "5", "2" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A quadratic function has its vertex at the point $(-9,9)$. The function passes through the point $(-7,-3)$. When written in vertex form, the function is $f(x)=a(x-h)^2+k$, where:\n$a=$ [ANS].\n$h=$ [ANS].\n$k=$ [ANS].", "answer_v2": [ "-3", "-9", "9" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A quadratic function has its vertex at the point $(-4,2)$. The function passes through the point $(-5,1)$. When written in vertex form, the function is $f(x)=a(x-h)^2+k$, where:\n$a=$ [ANS].\n$h=$ [ANS].\n$k=$ [ANS].", "answer_v3": [ "-1", "-4", "2" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0353", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Forms: vertex, factored, general", "level": "2", "keywords": [ "algebra", "function", "vertex" ], "problem_v1": "Find a function whose graph is a parabola with vertex (2,3) and that passes through the point (3,6). Your answer is ([ANS].", "answer_v1": [ "3*((x-2)*(x-2)+1)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find a function whose graph is a parabola with vertex (-4,12) and that passes through the point (-3,15). Your answer is ([ANS].", "answer_v2": [ "3*((x--4)*(x--4)+4)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find a function whose graph is a parabola with vertex (-2,3) and that passes through the point (-1,6). Your answer is ([ANS].", "answer_v3": [ "3*((x--2)*(x--2)+1)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0354", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Forms: vertex, factored, general", "level": "3", "keywords": [ "quadratic functions" ], "problem_v1": "Write the equation $-5 (4x-5) (x-1)=0$ in the standard form $a x^2+b x+c=0$. [ANS] $=0.$", "answer_v1": [ "45*x-20*x^2-25" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Write the equation $-2 (6x-2) (x-1)=0$ in the standard form $a x^2+b x+c=0$. [ANS] $=0.$", "answer_v2": [ "16*x-12*x^2-4" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Write the equation $-3 (5x-3) (x-1)=0$ in the standard form $a x^2+b x+c=0$. [ANS] $=0.$", "answer_v3": [ "24*x-15*x^2-9" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0355", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Inequalities", "level": "3", "keywords": [ "Algebra", "Inequalities" ], "problem_v1": "Using, the solution to the inequality\nx^2+5x-50 < 0 is all $x$ in the interval [ANS]. Note: This is equivalent to asking where the graph of the parabola $y=x^2+5x-50$ is below the $x$-axis.", "answer_v1": [ "(-10,5)" ], "answer_type_v1": [ "INT" ], "options_v1": [ [] ], "problem_v2": "Using, the solution to the inequality\nx^2+9x-22 < 0 is all $x$ in the interval [ANS]. Note: This is equivalent to asking where the graph of the parabola $y=x^2+9x-22$ is below the $x$-axis.", "answer_v2": [ "(-11,2)" ], "answer_type_v2": [ "INT" ], "options_v2": [ [] ], "problem_v3": "Using, the solution to the inequality\nx^2+7x-30 < 0 is all $x$ in the interval [ANS]. Note: This is equivalent to asking where the graph of the parabola $y=x^2+7x-30$ is below the $x$-axis.", "answer_v3": [ "(-10,3)" ], "answer_type_v3": [ "INT" ], "options_v3": [ [] ] }, { "id": "Algebra_0356", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Inequalities", "level": "3", "keywords": [ "polynomial", "inequality", "inequalities" ], "problem_v1": "Solve the following inequality and write your answer using interval notation.\nu^2+13 u+22 \\leq 3 u+1 Answer: [ANS]\nHelp:", "answer_v1": [ "[-7,-3]" ], "answer_type_v1": [ "INT" ], "options_v1": [ [] ], "problem_v2": "Solve the following inequality and write your answer using interval notation.\nx^2+20x+68 < 3x-2 Answer: [ANS]\nHelp:", "answer_v2": [ "(-10,-7)" ], "answer_type_v2": [ "INT" ], "options_v2": [ [] ], "problem_v3": "Solve the following inequality and write your answer using interval notation.\ny^2+6 y+19 \\geq-4 y-2 Answer: [ANS]\nHelp:", "answer_v3": [ "(-infinity,-7] U [-3,infinity)" ], "answer_type_v3": [ "INT" ], "options_v3": [ [] ] }, { "id": "Algebra_0357", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Graphs", "level": "2", "keywords": [ "quadratic", "function", "parabola", "vertex", "axis", "fraction" ], "problem_v1": "Find the axis of symmetry and vertex of the quadratic function ${y}={3x^{2}+3x+1}$. Use fractions in your answers.\nAxis of symmetry: [ANS]\nVertex: [ANS]", "answer_v1": [ "x = -1/2", "(-1/2,1/4)" ], "answer_type_v1": [ "EQ", "OL" ], "options_v1": [ [], [] ], "problem_v2": "Find the axis of symmetry and vertex of the quadratic function ${y}={-5x^{2}-25x-4}$. Use fractions in your answers.\nAxis of symmetry: [ANS]\nVertex: [ANS]", "answer_v2": [ "x = -5/2", "(-5/2,109/4)" ], "answer_type_v2": [ "EQ", "OL" ], "options_v2": [ [], [] ], "problem_v3": "Find the axis of symmetry and vertex of the quadratic function ${y}={-2x^{2}-2x-2}$. Use fractions in your answers.\nAxis of symmetry: [ANS]\nVertex: [ANS]", "answer_v3": [ "x = -1/2", "(-1/2,-3/2)" ], "answer_type_v3": [ "EQ", "OL" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0358", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Graphs", "level": "2", "keywords": [ "algebra" ], "problem_v1": "Find the intercepts and range of the function $f(x)=16x^2-13$.\nInstructions: Type the x-intercept with the smaller x-coordinate first. Type+INF for infinity, and-INF for negative infinity. \nx-intercepts: ([ANS], [ANS]) and ([ANS], [ANS]) y-intercept: ([ANS], [ANS]) range: [[ANS], [ANS])", "answer_v1": [ "-0.901387818865997", "0", "0.901387818865997", "0", "0", "-13", "-13", "+INF" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Find the intercepts and range of the function $f(x)=3x^2-19$.\nInstructions: Type the x-intercept with the smaller x-coordinate first. Type+INF for infinity, and-INF for negative infinity. \nx-intercepts: ([ANS], [ANS]) and ([ANS], [ANS]) y-intercept: ([ANS], [ANS]) range: [[ANS], [ANS])", "answer_v2": [ "-2.51661147842358", "0", "2.51661147842358", "0", "0", "-19", "-19", "+INF" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Find the intercepts and range of the function $f(x)=7x^2-13$.\nInstructions: Type the x-intercept with the smaller x-coordinate first. Type+INF for infinity, and-INF for negative infinity. \nx-intercepts: ([ANS], [ANS]) and ([ANS], [ANS]) y-intercept: ([ANS], [ANS]) range: [[ANS], [ANS])", "answer_v3": [ "-1.36277028773849", "0", "1.36277028773849", "0", "0", "-13", "-13", "+INF" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Algebra_0359", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "quadratic", "function", "parabola", "application" ], "problem_v1": "One number is $9$ less than a second number. Find a pair of such number that their product is as small as possible.\nThese two numbers are [ANS]. (Use a comma to separate your numbers.) The smallest possible product is [ANS].", "answer_v1": [ "(-4.5, 4.5)", "-20.25" ], "answer_type_v1": [ "UOL", "NV" ], "options_v1": [ [], [] ], "problem_v2": "One number is $4$ less than a second number. Find a pair of such number that their product is as small as possible.\nThese two numbers are [ANS]. (Use a comma to separate your numbers.) The smallest possible product is [ANS].", "answer_v2": [ "(-2, 2)", "-4" ], "answer_type_v2": [ "UOL", "NV" ], "options_v2": [ [], [] ], "problem_v3": "One number is $6$ less than a second number. Find a pair of such number that their product is as small as possible.\nThese two numbers are [ANS]. (Use a comma to separate your numbers.) The smallest possible product is [ANS].", "answer_v3": [ "(-3, 3)", "-9" ], "answer_type_v3": [ "UOL", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0360", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "quadratic", "function", "parabola", "application" ], "problem_v1": "You plan to build four identical rectangular sheep pens in a row. Each adjacent pair of pens share a fence between them. You have a total of $400$ feet of fence to use. Find the dimension of each pen such that you can enclose the maximum area.\nThe length of each pen (along the walls that they share) should be [ANS]. The width of each pen should be [ANS]. The maximum area of each pen is [ANS]. (Use ft for feet, and ft^2 for square feet.)", "answer_v1": [ "40", "25", "1000" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "You plan to build four identical rectangular sheep pens in a row. Each adjacent pair of pens share a fence between them. You have a total of $312$ feet of fence to use. Find the dimension of each pen such that you can enclose the maximum area.\nThe length of each pen (along the walls that they share) should be [ANS]. The width of each pen should be [ANS]. The maximum area of each pen is [ANS]. (Use ft for feet, and ft^2 for square feet.)", "answer_v2": [ "31.2", "19.5", "608.4" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "You plan to build four identical rectangular sheep pens in a row. Each adjacent pair of pens share a fence between them. You have a total of $344$ feet of fence to use. Find the dimension of each pen such that you can enclose the maximum area.\nThe length of each pen (along the walls that they share) should be [ANS]. The width of each pen should be [ANS]. The maximum area of each pen is [ANS]. (Use ft for feet, and ft^2 for square feet.)", "answer_v3": [ "34.4", "21.5", "739.6" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0361", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "quadratic", "function", "parabola", "application" ], "problem_v1": "Currently, an artist can sell $240$ paintings every year at the price of ${\\$100.00}$ per painting. Each time he raises the price per painting by ${\\$5.00}$, he sells $5$ fewer paintings every year. Assume the artist will raise the price per painting $x$ times. The current price per painting is ${\\$100.00}$. After raising the price $x$ times, each time by ${\\$5.00}$, the new price per painting will become $100+5x$ dollars. Currently he sells $240$ paintings per year. It\u2019s given that he will sell $5$ fewer paintings each time he raises the price. After raising the price per painting $x$ times, he will sell $240-5x$ paintings every year. The artist\u2019s income can be calculated by multiplying the number of paintings sold with price per painting. If he raises the price per painting $x$ times, his new yearly income can be modeled by the function:\n${f(x)=(100+5x)(240-5x)}$ where $f(x)$ stands for his yearly income in dollars. Answer the following questions: 1) To obtain maximum income of [ANS], the artist should set the price per painting at [ANS]. 2) To earn ${\\$27{,}675.00}$ per year, the artist could sell his paintings at two different prices. The lower price is [ANS] per painting, and the higher price is [ANS] per painting.", "answer_v1": [ "28900.00", "170.00", "135.00", "205.00" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Currently, an artist can sell $210$ paintings every year at the price of ${\\$120.00}$ per painting. Each time he raises the price per painting by ${\\$15.00}$, he sells $10$ fewer paintings every year. Assume the artist will raise the price per painting $x$ times. The current price per painting is ${\\$120.00}$. After raising the price $x$ times, each time by ${\\$15.00}$, the new price per painting will become $120+15x$ dollars. Currently he sells $210$ paintings per year. It\u2019s given that he will sell $10$ fewer paintings each time he raises the price. After raising the price per painting $x$ times, he will sell $210-10x$ paintings every year. The artist\u2019s income can be calculated by multiplying the number of paintings sold with price per painting. If he raises the price per painting $x$ times, his new yearly income can be modeled by the function:\n${f(x)=(120+15x)(210-10x)}$ where $f(x)$ stands for his yearly income in dollars. Answer the following questions: 1) To obtain maximum income of [ANS], the artist should set the price per painting at [ANS]. 2) To earn ${\\$30{,}600.00}$ per year, the artist could sell his paintings at two different prices. The lower price is [ANS] per painting, and the higher price is [ANS] per painting.", "answer_v2": [ "31537.50", "$217.50", "$180.00", "$255.00" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Currently, an artist can sell $280$ paintings every year at the price of ${\\$60.00}$ per painting. Each time he raises the price per painting by ${\\$10.00}$, he sells $5$ fewer paintings every year. Assume the artist will raise the price per painting $x$ times. The current price per painting is ${\\$60.00}$. After raising the price $x$ times, each time by ${\\$10.00}$, the new price per painting will become $60+10x$ dollars. Currently he sells $280$ paintings per year. It\u2019s given that he will sell $5$ fewer paintings each time he raises the price. After raising the price per painting $x$ times, he will sell $280-5x$ paintings every year. The artist\u2019s income can be calculated by multiplying the number of paintings sold with price per painting. If he raises the price per painting $x$ times, his new yearly income can be modeled by the function:\n${f(x)=(60+10x)(280-5x)}$ where $f(x)$ stands for his yearly income in dollars. Answer the following questions: 1) To obtain maximum income of [ANS], the artist should set the price per painting at [ANS]. 2) To earn ${\\$38{,}250.00}$ per year, the artist could sell his paintings at two different prices. The lower price is [ANS] per painting, and the higher price is [ANS] per painting.", "answer_v3": [ "48050.00", "$310.00", "$170.00", "$450.00" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Algebra_0362", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "algebra", "word problem' 'cutting wire", "quadratic" ], "problem_v1": "You have a wire that is 80 cm long. You wish to cut it into two pieces. One piece will be bent into the shape of a square. The other piece will be bent into the shape of a circle. Let A represent the total area of the square and the circle. What is the circumference of the circle when A is a minimum?\nThe circumference of the circle is [ANS] cm.", "answer_v1": [ "35.1920677190754" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "You have a wire that is 26 cm long. You wish to cut it into two pieces. One piece will be bent into the shape of a square. The other piece will be bent into the shape of a circle. Let A represent the total area of the square and the circle. What is the circumference of the circle when A is a minimum?\nThe circumference of the circle is [ANS] cm.", "answer_v2": [ "11.4374220086995" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "You have a wire that is 44 cm long. You wish to cut it into two pieces. One piece will be bent into the shape of a square. The other piece will be bent into the shape of a circle. Let A represent the total area of the square and the circle. What is the circumference of the circle when A is a minimum?\nThe circumference of the circle is [ANS] cm.", "answer_v3": [ "19.3556372454915" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0363", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "algebra", "word problem" ], "problem_v1": "A factory is to be built on a lot measuring 270 ft by 360 ft. A local building code specifies that a lawn of uniform width and equal in area to the factory must surround the factory.\nWhat must the width of the lawn be? [ANS]\nIf the dimensions of the factory are $A$ ft by $B$ ft with $A < B$, then $A=$ [ANS] and $B=$ [ANS]", "answer_v1": [ "45", "180", "270" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A factory is to be built on a lot measuring 150 ft by 200 ft. A local building code specifies that a lawn of uniform width and equal in area to the factory must surround the factory.\nWhat must the width of the lawn be? [ANS]\nIf the dimensions of the factory are $A$ ft by $B$ ft with $A < B$, then $A=$ [ANS] and $B=$ [ANS]", "answer_v2": [ "25", "100", "150" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A factory is to be built on a lot measuring 180 ft by 240 ft. A local building code specifies that a lawn of uniform width and equal in area to the factory must surround the factory.\nWhat must the width of the lawn be? [ANS]\nIf the dimensions of the factory are $A$ ft by $B$ ft with $A < B$, then $A=$ [ANS] and $B=$ [ANS]", "answer_v3": [ "30", "120", "180" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0364", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Applications and models", "level": "2", "keywords": [ "Algebra", "Modeling" ], "problem_v1": "A Norman window has the shape of a rectangle surmounted by a semicircle. The perimeter is 43.000 ft. Order the widths listed below according to the area of the corresponding Norman window from the lowest area (1) to highest area (5). You will need to enter the numbers 1 through 5 in the entry blanks below. [ANS] 1. Width=9.600 ft. [ANS] 2. Width=10.000 ft. [ANS] 3. Width=11.200 ft. [ANS] 4. Width=6.400 ft. [ANS] 5. Width=6.700 ft.\nRemark: To be able to order the sizes of the windows you are going to have to calculate the area for all five windows from knowing their widths. Since there are several calculations it will save time to figure out and simplify a formula which calculates the area from the width and the perimeter.", "answer_v1": [ "3", "4", "5", "1", "2" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "A Norman window has the shape of a rectangle surmounted by a semicircle. The perimeter is 22.000 ft. Order the widths listed below according to the area of the corresponding Norman window from the lowest area (1) to highest area (5). You will need to enter the numbers 1 through 5 in the entry blanks below. [ANS] 1. Width=7.000 ft. [ANS] 2. Width=3.600 ft. [ANS] 3. Width=4.400 ft. [ANS] 4. Width=7.100 ft. [ANS] 5. Width=4.300 ft.\nRemark: To be able to order the sizes of the windows you are going to have to calculate the area for all five windows from knowing their widths. Since there are several calculations it will save time to figure out and simplify a formula which calculates the area from the width and the perimeter.", "answer_v2": [ "5", "1", "3", "4", "2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "A Norman window has the shape of a rectangle surmounted by a semicircle. The perimeter is 29.000 ft. Order the widths listed below according to the area of the corresponding Norman window from the lowest area (1) to highest area (5). You will need to enter the numbers 1 through 5 in the entry blanks below. [ANS] 1. Width=7.000 ft. [ANS] 2. Width=4.800 ft. [ANS] 3. Width=6.600 ft. [ANS] 4. Width=4.300 ft. [ANS] 5. Width=5.300 ft.\nRemark: To be able to order the sizes of the windows you are going to have to calculate the area for all five windows from knowing their widths. Since there are several calculations it will save time to figure out and simplify a formula which calculates the area from the width and the perimeter.", "answer_v3": [ "5", "2", "4", "1", "3" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Algebra_0365", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "Algebra", "Modeling" ], "problem_v1": "Taxylvania has a tax code that rewards charitable giving. If a person gives $p$ \\% of his income to charity, that person pays $(43-1.6 p)$ \\% tax on the remaining money. For example, if a person gives 10\\% of his income to charity, he pays 27 \\% tax on the remaining money. If a person gives 26.875 \\% of his income to charity, he pays no tax on the remaining money. A person does not receive a tax refund if he gives more than 26.875 \\% of his income to charity. Count Taxula earns \\$ 60000. What percentage of his income should he give to charity to maximize the money he has after taxes and charitable giving?\nThe count should give [ANS] \\% to charity.\nIf the count did receive a tax refund for giving more than 26.875 \\% of his income to charity, how much should he give to charity?\nThe count should give [ANS] \\% to charity.\nNOTE: Your answers must be numbers. No arithmetic operations are allowed.", "answer_v1": [ "26.875", "32.1875" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Taxylvania has a tax code that rewards charitable giving. If a person gives $p$ \\% of his income to charity, that person pays $(22-1.9 p)$ \\% tax on the remaining money. For example, if a person gives 10\\% of his income to charity, he pays 3 \\% tax on the remaining money. If a person gives 11.5789473684211 \\% of his income to charity, he pays no tax on the remaining money. A person does not receive a tax refund if he gives more than 11.5789473684211 \\% of his income to charity. Count Taxula earns \\$ 22000. What percentage of his income should he give to charity to maximize the money he has after taxes and charitable giving?\nThe count should give [ANS] \\% to charity.\nIf the count did receive a tax refund for giving more than 11.5789473684211 \\% of his income to charity, how much should he give to charity?\nThe count should give [ANS] \\% to charity.\nNOTE: Your answers must be numbers. No arithmetic operations are allowed.", "answer_v2": [ "11.5789473684211", "29.4736842105263" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Taxylvania has a tax code that rewards charitable giving. If a person gives $p$ \\% of his income to charity, that person pays $(29-1.6 p)$ \\% tax on the remaining money. For example, if a person gives 10\\% of his income to charity, he pays 13 \\% tax on the remaining money. If a person gives 18.125 \\% of his income to charity, he pays no tax on the remaining money. A person does not receive a tax refund if he gives more than 18.125 \\% of his income to charity. Count Taxula earns \\$ 32000. What percentage of his income should he give to charity to maximize the money he has after taxes and charitable giving?\nThe count should give [ANS] \\% to charity.\nIf the count did receive a tax refund for giving more than 18.125 \\% of his income to charity, how much should he give to charity?\nThe count should give [ANS] \\% to charity.\nNOTE: Your answers must be numbers. No arithmetic operations are allowed.", "answer_v3": [ "18.125", "27.8125" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0366", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "algebra", "graph" ], "problem_v1": "Find the area of the region that lies outside the circle x^2+y^2=4 but inside the circle x^2+y^2-4y-32=0. The area is [ANS] square units", "answer_v1": [ "32*pi" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the area of the region that lies outside the circle x^2+y^2=1 but inside the circle x^2+y^2-4y-60=0. The area is [ANS] square units", "answer_v2": [ "63*pi" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the area of the region that lies outside the circle x^2+y^2=1 but inside the circle x^2+y^2-4y-45=0. The area is [ANS] square units", "answer_v3": [ "48*pi" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0367", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "algebra", "word problem", "quadratic" ], "problem_v1": "Given that the surface area of a sphere is 163 $\\pi$ cm $^2$, find its volume.\nNote: Your answer must be a number. No arithmetic operations are allowed.\nThe volume of the sphere is [ANS] cm $^3$.", "answer_v1": [ "1089.63245142577" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Given that the surface area of a sphere is 35 $\\pi$ cm $^2$, find its volume.\nNote: Your answer must be a number. No arithmetic operations are allowed.\nThe volume of the sphere is [ANS] cm $^3$.", "answer_v2": [ "108.417824577048" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Given that the surface area of a sphere is 79 $\\pi$ cm $^2$, find its volume.\nNote: Your answer must be a number. No arithmetic operations are allowed.\nThe volume of the sphere is [ANS] cm $^3$.", "answer_v3": [ "367.6539694207" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0368", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "calculus", "derivative", "polynomials", "powers", "functions", "rational functions" ], "problem_v1": "According to Car and Driver, an Alfa Romeo going 70 mph requires 177 feet to stop. Assuming that the stopping distance is proportional to the square of the velocity, find the stopping distance required by an Alfa Romeo going at 55 mph and at 120 mph. At 55 mph, stopping distance=[ANS] (include) At 120 mph, stopping distance=[ANS] (include)", "answer_v1": [ "109.27", "520.163" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "According to Car and Driver, an Alfa Romeo going 70 mph requires 177 feet to stop. Assuming that the stopping distance is proportional to the square of the velocity, find the stopping distance required by an Alfa Romeo going at 15 mph and at 140 mph. At 15 mph, stopping distance=[ANS] (include) At 140 mph, stopping distance=[ANS] (include)", "answer_v2": [ "8.12755", "708" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "According to Car and Driver, an Alfa Romeo going 70 mph requires 177 feet to stop. Assuming that the stopping distance is proportional to the square of the velocity, find the stopping distance required by an Alfa Romeo going at 30 mph and at 120 mph. At 30 mph, stopping distance=[ANS] (include) At 120 mph, stopping distance=[ANS] (include)", "answer_v3": [ "32.5102", "520.163" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0369", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [], "problem_v1": "A book is opened to a page at random. The product of the facing page numbers is 2970. The sum of the facing page numbers is [ANS]", "answer_v1": [ "109" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A book is opened to a page at random. The product of the facing page numbers is 1806. The sum of the facing page numbers is [ANS]", "answer_v2": [ "85" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A book is opened to a page at random. The product of the facing page numbers is 2162. The sum of the facing page numbers is [ANS]", "answer_v3": [ "93" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0370", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "algebra" ], "problem_v1": "You are on a pleasure cruise through the universe and you crash in the ocean of an unknown planet. Your spaceship floats on the water and its top is 32 feet above the surface of the water. You swim away from the spaceship until you see its top on the horizon. Your laser range meter tells you that your eyes are 2.8 miles away from the top of the space ship. (You are a capable-if reckless and curious-swimmer.) The radius of the planet is [ANS] miles. It's a small world, but it's all yours. You figure that the surface of the earth is [ANS] times as large as the surface of your planet, but still, your planet is plenty big enough for you (if only you can find land somewhere). Hint: This is like the preceding problem except that the unknown is different.", "answer_v1": [ "646.79696969697", "37.5415493069361" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "You are on a pleasure cruise through the universe and you crash in the ocean of an unknown planet. Your spaceship floats on the water and its top is 39 feet above the surface of the water. You swim away from the spaceship until you see its top on the horizon. Your laser range meter tells you that your eyes are 2 miles away from the top of the space ship. (You are a capable-if reckless and curious-swimmer.) The radius of the planet is [ANS] miles. It's a small world, but it's all yours. You figure that the surface of the earth is [ANS] times as large as the surface of your planet, but still, your planet is plenty big enough for you (if only you can find land somewhere). Hint: This is like the preceding problem except that the unknown is different.", "answer_v2": [ "270.765537587413", "214.220668186019" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "You are on a pleasure cruise through the universe and you crash in the ocean of an unknown planet. Your spaceship floats on the water and its top is 32 feet above the surface of the water. You swim away from the spaceship until you see its top on the horizon. Your laser range meter tells you that your eyes are 2.3 miles away from the top of the space ship. (You are a capable-if reckless and curious-swimmer.) The radius of the planet is [ANS] miles. It's a small world, but it's all yours. You figure that the surface of the earth is [ANS] times as large as the surface of your planet, but still, your planet is plenty big enough for you (if only you can find land somewhere). Hint: This is like the preceding problem except that the unknown is different.", "answer_v3": [ "436.42196969697", "82.4584058710387" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0371", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "algebra" ], "problem_v1": "You are flying in an open plane at an altitude of 7744 feet and you drop a Coca Cola bottle out of the window. The bottle will hit the ground after [ANS] seconds. (Note: this problem is a bit unrealistic since it ignores air drag. The scenario described here provides the opening scene in the movie \"The Gods Must Be Crazy.\")", "answer_v1": [ "22" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "You are flying in an open plane at an altitude of 2304 feet and you drop a Coca Cola bottle out of the window. The bottle will hit the ground after [ANS] seconds. (Note: this problem is a bit unrealistic since it ignores air drag. The scenario described here provides the opening scene in the movie \"The Gods Must Be Crazy.\")", "answer_v2": [ "12" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "You are flying in an open plane at an altitude of 3600 feet and you drop a Coca Cola bottle out of the window. The bottle will hit the ground after [ANS] seconds. (Note: this problem is a bit unrealistic since it ignores air drag. The scenario described here provides the opening scene in the movie \"The Gods Must Be Crazy.\")", "answer_v3": [ "15" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0372", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "algebra" ], "problem_v1": "You fire a rifle at an angle of 45 degrees. Thus the initial horizontal and vertical velocities of your bullet are the same. Suppose they each equal 368 feet per second. Again ignore air resistance. Assume you are shooting from ground level (height 0). Your bullet will hit the ground [ANS] feet from your current position.", "answer_v1": [ "8464" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "You fire a rifle at an angle of 45 degrees. Thus the initial horizontal and vertical velocities of your bullet are the same. Suppose they each equal 240 feet per second. Again ignore air resistance. Assume you are shooting from ground level (height 0). Your bullet will hit the ground [ANS] feet from your current position.", "answer_v2": [ "3600" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "You fire a rifle at an angle of 45 degrees. Thus the initial horizontal and vertical velocities of your bullet are the same. Suppose they each equal 288 feet per second. Again ignore air resistance. Assume you are shooting from ground level (height 0). Your bullet will hit the ground [ANS] feet from your current position.", "answer_v3": [ "5184" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0373", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "algebra", "calculus" ], "problem_v1": "You drop a rock into a deep well. You can't see the rock's impact at the bottom, but you hear it after 7 seconds. The depth of the well is [ANS] feet. Ignore air resistance. The time that passes after you drop the rock has two components: the time it takes the rock to reach the bottom of the well, and the time that it takes the sound of the impact to travel back to you. Assume the speed of sound is 1100 feet per second. Note: After $t$ seconds the rock has reached a depth of $d$ feet where d=16t^2. Set up and solve a quadratic equation.", "answer_v1": [ "656.088459953688" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "You drop a rock into a deep well. You can't see the rock's impact at the bottom, but you hear it after 4 seconds. The depth of the well is [ANS] feet. Ignore air resistance. The time that passes after you drop the rock has two components: the time it takes the rock to reach the bottom of the well, and the time that it takes the sound of the impact to travel back to you. Assume the speed of sound is 1100 feet per second. Note: After $t$ seconds the rock has reached a depth of $d$ feet where d=16t^2. Set up and solve a quadratic equation.", "answer_v2": [ "229.942237876931" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "You drop a rock into a deep well. You can't see the rock's impact at the bottom, but you hear it after 5 seconds. The depth of the well is [ANS] feet. Ignore air resistance. The time that passes after you drop the rock has two components: the time it takes the rock to reach the bottom of the well, and the time that it takes the sound of the impact to travel back to you. Assume the speed of sound is 1100 feet per second. Note: After $t$ seconds the rock has reached a depth of $d$ feet where d=16t^2. Set up and solve a quadratic equation.", "answer_v3": [ "350.625550088018" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0374", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "algebra" ], "problem_v1": "A rectangular plot, 19 ft by 18 ft, is to be used for a garden. It is decided to put a pavement inside the entire border so that 110 square feet of the plot is left for flowers. How wide should the pavement be? Answer: [ANS] feet", "answer_v1": [ "4" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A rectangular plot, 15 ft by 20 ft, is to be used for a garden. It is decided to put a pavement inside the entire border so that 176 square feet of the plot is left for flowers. How wide should the pavement be? Answer: [ANS] feet", "answer_v2": [ "2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A rectangular plot, 16 ft by 18 ft, is to be used for a garden. It is decided to put a pavement inside the entire border so that 120 square feet of the plot is left for flowers. How wide should the pavement be? Answer: [ANS] feet", "answer_v3": [ "3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0375", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "algebra" ], "problem_v1": "The demand function for a product is $p=112-7 q$ where $p$ is the price in dollars when $q$ units are demanded. Find the level of production that maximizes the total revenue and determine the revenue. $q=$ [ANS] units $R=$ \\$ [ANS]", "answer_v1": [ "8", "448" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The demand function for a product is $p=40-10 q$ where $p$ is the price in dollars when $q$ units are demanded. Find the level of production that maximizes the total revenue and determine the revenue. $q=$ [ANS] units $R=$ \\$ [ANS]", "answer_v2": [ "2", "40" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The demand function for a product is $p=56-7 q$ where $p$ is the price in dollars when $q$ units are demanded. Find the level of production that maximizes the total revenue and determine the revenue. $q=$ [ANS] units $R=$ \\$ [ANS]", "answer_v3": [ "4", "112" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0376", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "power equations" ], "problem_v1": "The volume of a cone of height $4$ and radius $r$ is $ V=\\frac{4}{3} \\pi r^2$. What is the radius of such a cone whose volume is $5 \\pi$?\n$r$=[ANS]", "answer_v1": [ "sqrt(3*5/4)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The volume of a cone of height $6$ and radius $r$ is $ V=\\frac{6}{3} \\pi r^2$. What is the radius of such a cone whose volume is $2 \\pi$?\n$r$=[ANS]", "answer_v2": [ "sqrt(3*2/6)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The volume of a cone of height $5$ and radius $r$ is $ V=\\frac{5}{3} \\pi r^2$. What is the radius of such a cone whose volume is $3 \\pi$?\n$r$=[ANS]", "answer_v3": [ "sqrt(3*3/5)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0377", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "quatratic inequality" ], "problem_v1": "Suppose $C(x)=8x^{2}-56x+115$, $x \\ge 0$ represents the cost, in hundreds of dollars, to produce $x$ thousands of pens. Find the number of pens which can be produced for no more than \\$ $1900$. Answer: between [ANS] thousand and [ANS] thousand pens", "answer_v1": [ "3", "4" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose $C(x)=5x^{2}-30x+37$, $x \\ge 0$ represents the cost, in hundreds of dollars, to produce $x$ thousands of pens. Find the number of pens which can be produced for no more than \\$ $1200$. Answer: between [ANS] thousand and [ANS] thousand pens", "answer_v2": [ "1", "5" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose $C(x)=7x^{2}-35x+42$, $x \\ge 0$ represents the cost, in hundreds of dollars, to produce $x$ thousands of pens. Find the number of pens which can be produced for no more than \\$ $1400$. Answer: between [ANS] thousand and [ANS] thousand pens", "answer_v3": [ "1", "4" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0378", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "equations", "quadratic" ], "problem_v1": "The product of two consecutive positive even integers is $440$. Find the integers. Answer: [ANS]\nHelp: Separate the integers by a comma.", "answer_v1": [ "(20, 22)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "The product of two consecutive positive odd integers is $143$. Find the integers. Answer: [ANS]\nHelp: Separate the integers by a comma.", "answer_v2": [ "(11, 13)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "The product of two consecutive positive odd integers is $255$. Find the integers. Answer: [ANS]\nHelp: Separate the integers by a comma.", "answer_v3": [ "(15, 17)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Algebra_0379", "subject": "Algebra", "topic": "Quadratic equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "algebra", "factoring" ], "problem_v1": "A room contains $78$ chairs arranged in rows. The number of rows is one more than twice the number of chairs per row. Find the number of rows. Number of Rows: [ANS]", "answer_v1": [ "2*6+1" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A room contains $21$ chairs arranged in rows. The number of rows is one more than twice the number of chairs per row. Find the number of rows. Number of Rows: [ANS]", "answer_v2": [ "2*3+1" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A room contains $36$ chairs arranged in rows. The number of rows is one more than twice the number of chairs per row. Find the number of rows. Number of Rows: [ANS]", "answer_v3": [ "2*4+1" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0380", "subject": "Algebra", "topic": "Operations on polynomial and rational expressions", "subtopic": "Polynomials: add, subtract", "level": "2", "keywords": [ "polynomial", "addition", "simplify", "binomial", "combine", "like terms" ], "problem_v1": "Use a vertical format to add the polynomials.\n$\\begin{alignedat}{2} 16x^{7}&{}-{}&9x^{5}\\\\ 13x^{7}&{}+{}&15x^{5}\\\\ \\hline \\end{alignedat}$ [ANS]", "answer_v1": [ "29*x^7+6*x^5" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Use a vertical format to add the polynomials.\n$\\begin{alignedat}{2} 2x^{19}&{}-{}&3x^{13}\\\\ 3x^{19}&{}+{}&7x^{13}\\\\ \\hline \\end{alignedat}$ [ANS]", "answer_v2": [ "5*x^19+4*x^13" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Use a vertical format to add the polynomials.\n$\\begin{alignedat}{2} 7x^{5}&{}-{}&9x^{3}\\\\ 6x^{5}&{}+{}&11x^{3}\\\\ \\hline \\end{alignedat}$ [ANS]", "answer_v3": [ "13*x^5+2*x^3" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0381", "subject": "Algebra", "topic": "Operations on polynomial and rational expressions", "subtopic": "Polynomials: add, subtract", "level": "2", "keywords": [ "polynomial", "add", "simplify", "binomial", "combine", "like terms" ], "problem_v1": "Add the two binomials, making sure to simplify your answer as much as possible.\n$\\left({5x+2}\\right)+\\left({3x+5}\\right)$ [ANS]", "answer_v1": [ "8*x+7" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Add the two binomials, making sure to simplify your answer as much as possible.\n$\\left({-9x+9}\\right)+\\left({-7x-3}\\right)$ [ANS]", "answer_v2": [ "-16*x+6" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Add the two binomials, making sure to simplify your answer as much as possible.\n$\\left({-4x+2}\\right)+\\left({-5x+1}\\right)$ [ANS]", "answer_v3": [ "-9*x+3" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0382", "subject": "Algebra", "topic": "Operations on polynomial and rational expressions", "subtopic": "Polynomials: add, subtract", "level": "2", "keywords": [ "polynomial", "trinomial", "simplify", "combine", "like terms", "subtract" ], "problem_v1": "Subtract $-5t^{14}-7t^{12}-6t^{10}$ from the sum of $6t^{14}-5t^{12}+8t^{10}$ and $-8t^{14}+4t^{12}-5t^{10}$. [ANS]", "answer_v1": [ "3*t^14+6*t^12+9*t^10" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Subtract $-8x^{13}-5x^{8}-10x^{7}$ from the sum of $10x^{13}-9x^{8}+4x^{7}$ and $-2x^{13}+4x^{8}-9x^{7}$. [ANS]", "answer_v2": [ "16*x^13+5*x^7" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Subtract $-2y^{6}-3y^{5}-9y^{4}$ from the sum of $7y^{6}-8y^{5}+6y^{4}$ and $-9y^{6}+4y^{5}-3y^{4}$. [ANS]", "answer_v3": [ "-y^5+12*y^4" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0383", "subject": "Algebra", "topic": "Operations on polynomial and rational expressions", "subtopic": "Polynomials: add, subtract", "level": "2", "keywords": [ "expanding", "polynomial", "polynomials" ], "problem_v1": "Perform the operation\n(-u^4-2 u^3-u^2-5 u-5)+(-2 u^3+4 u^2-u-3) and combine line terms. Simplify your answer as much as possible. Answer: [ANS]", "answer_v1": [ "-u^4-4*u^3+3*u^2-6*u-8" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Perform the operation\n(-5x^3-3x^2+x-6)+(6x^4-5x^3-2x^2+7x-3) and combine line terms. Simplify your answer as much as possible. Answer: [ANS]", "answer_v2": [ "6*x^4-10*x^3-5*x^2+8*x-9" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Perform the operation\n(2 z^4-3 z^3+z^2-4 z-2)+(5 z^3+6 z^2+6 z-4) and combine line terms. Simplify your answer as much as possible. Answer: [ANS]", "answer_v3": [ "2*z^4+2*z^3+7*z^2+2*z-6" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0384", "subject": "Algebra", "topic": "Operations on polynomial and rational expressions", "subtopic": "Polynomials: add, subtract", "level": "2", "keywords": [ "expanding", "polynomial", "polynomials" ], "problem_v1": "Expand the expression\n(8 p+4)^2 and combine like terms. Simplify your answer as much as possible. Answer: [ANS]", "answer_v1": [ "64*p^2+64*p+16" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Expand the expression\n(-12x-3)^2 and combine like terms. Simplify your answer as much as possible. Answer: [ANS]", "answer_v2": [ "144*x^2+72*x+9" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Expand the expression\n(-8 y+1)^2 and combine like terms. Simplify your answer as much as possible. Answer: [ANS]", "answer_v3": [ "64*y^2-16*y+1" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0385", "subject": "Algebra", "topic": "Operations on polynomial and rational expressions", "subtopic": "Polynomials: multiply", "level": "2", "keywords": [ "multivariable", "multiply", "polynomial", "FOIL", "difference of squares" ], "problem_v1": "Find the product\n${({a+b+8})({a+b-8})=}$ [ANS]", "answer_v1": [ "a^2+2*a*b+b^2-64" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the product\n${({a+b+2})({a+b-2})=}$ [ANS]", "answer_v2": [ "a^2+2*a*b+b^2-4" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the product\n${({a+b+4})({a+b-4})=}$ [ANS]", "answer_v3": [ "a^2+2*a*b+b^2-16" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0386", "subject": "Algebra", "topic": "Operations on polynomial and rational expressions", "subtopic": "Polynomials: multiply", "level": "3", "keywords": [ "algebra", "Word Problems", "Translations" ], "problem_v1": "A square rug lies in the middle of a rectangular room. There are 6 feet of uncovered floor on 2 sides of the rug and 5 feet of uncovered floor on the other 2 sides. Find a polynomial expression for the area of the room in terms of x, the side length of the rug.\nThe area of the room is [ANS].", "answer_v1": [ "(2*6 +x)*(2*5+x)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "A square rug lies in the middle of a rectangular room. There are 2 feet of uncovered floor on 2 sides of the rug and 7 feet of uncovered floor on the other 2 sides. Find a polynomial expression for the area of the room in terms of x, the side length of the rug.\nThe area of the room is [ANS].", "answer_v2": [ "(2*2 +x)*(2*7+x)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "A square rug lies in the middle of a rectangular room. There are 3 feet of uncovered floor on 2 sides of the rug and 5 feet of uncovered floor on the other 2 sides. Find a polynomial expression for the area of the room in terms of x, the side length of the rug.\nThe area of the room is [ANS].", "answer_v3": [ "(2*3 +x)*(2*5+x)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0387", "subject": "Algebra", "topic": "Operations on polynomial and rational expressions", "subtopic": "Polynomials: multiply", "level": "2", "keywords": [ "algebra", "polynomial arithmetic" ], "problem_v1": "Given P=7 b^3+2 b-6, Q=b^2+4 b+3, R=b^3-3 Then $P+Q=$ [ANS] $b^3+$ [ANS] $b^2+$ [ANS] $b+$ [ANS]\nand $R(P+Q)=$ [ANS] $b^6+$ [ANS] $b^5+$ [ANS] $b^4+$ [ANS] $b^3+$ [ANS] $b^2+$ [ANS] $b+$ [ANS]", "answer_v1": [ "7", "1", "6", "-3", "7", "1", "6", "-24", "-3", "-18", "9" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Given P=1 b^3+8 b-2, Q=b^2-3 b+9, R=b^3-3 Then $P+Q=$ [ANS] $b^3+$ [ANS] $b^2+$ [ANS] $b+$ [ANS]\nand $R(P+Q)=$ [ANS] $b^6+$ [ANS] $b^5+$ [ANS] $b^4+$ [ANS] $b^3+$ [ANS] $b^2+$ [ANS] $b+$ [ANS]", "answer_v2": [ "1", "1", "5", "7", "1", "1", "5", "4", "-3", "-15", "-21" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Given P=3 b^3+2 b-3, Q=b^2+1 b+2, R=b^3-3 Then $P+Q=$ [ANS] $b^3+$ [ANS] $b^2+$ [ANS] $b+$ [ANS]\nand $R(P+Q)=$ [ANS] $b^6+$ [ANS] $b^5+$ [ANS] $b^4+$ [ANS] $b^3+$ [ANS] $b^2+$ [ANS] $b+$ [ANS]", "answer_v3": [ "3", "1", "3", "-1", "3", "1", "3", "-10", "-3", "-9", "3" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Algebra_0388", "subject": "Algebra", "topic": "Operations on polynomial and rational expressions", "subtopic": "Polynomials: multiply", "level": "2", "keywords": [ "algebra", "polynomials" ], "problem_v1": "Find the indicated product. (3x-7)(3x+7) Answer: [ANS]", "answer_v1": [ "9*x^2+-49" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the indicated product. (2x-10)(2x+10) Answer: [ANS]", "answer_v2": [ "4*x^2+-100" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the indicated product. (2x-7)(2x+7) Answer: [ANS]", "answer_v3": [ "4*x^2+-49" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0389", "subject": "Algebra", "topic": "Operations on polynomial and rational expressions", "subtopic": "Polynomials: divide", "level": "2", "keywords": [ "algebra", "remainder theorem", "synthetic division" ], "problem_v1": "Use long division and the Remainder Theorem to evaluate $P(c)$, where P(x)=x^3-9x^2+19x-15, \\quad c=2. The quotient is [ANS]\nThe remainder is [ANS]\n$P(c)$=[ANS]", "answer_v1": [ "x**2-7*x+5", "-5", "-5" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Use long division and the Remainder Theorem to evaluate $P(c)$, where P(x)=x^3-3x^2+10x-18, \\quad c=2. The quotient is [ANS]\nThe remainder is [ANS]\n$P(c)$=[ANS]", "answer_v2": [ "x**2-1*x+8", "-2", "-2" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Use long division and the Remainder Theorem to evaluate $P(c)$, where P(x)=x^3-5x^2+11x-13, \\quad c=2. The quotient is [ANS]\nThe remainder is [ANS]\n$P(c)$=[ANS]", "answer_v3": [ "x**2-3*x+5", "-3", "-3" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0390", "subject": "Algebra", "topic": "Operations on polynomial and rational expressions", "subtopic": "Polynomials: divide", "level": "2", "keywords": [ "algebra", "synthetic division" ], "problem_v1": "Find the quotient and remainder using synthetic division for \\frac{x^5-x^4+7x^3-7x^2+5x-10}{x-1}. The quotient is [ANS]\nThe remainder is [ANS]", "answer_v1": [ "x**4+7*x**2+5", "-5" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the quotient and remainder using synthetic division for \\frac{x^5-x^4+1x^3-1x^2+8x-10}{x-1}. The quotient is [ANS]\nThe remainder is [ANS]", "answer_v2": [ "x**4+1*x**2+8", "-2" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the quotient and remainder using synthetic division for \\frac{x^5-x^4+3x^3-3x^2+5x-8}{x-1}. The quotient is [ANS]\nThe remainder is [ANS]", "answer_v3": [ "x**4+3*x**2+5", "-3" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0391", "subject": "Algebra", "topic": "Operations on polynomial and rational expressions", "subtopic": "Polynomials: divide", "level": "3", "keywords": [ "algebra", "polynomials", "division" ], "problem_v1": "Find $k$ such that the Polynomial $P(x)=4x^3+k x^2+4x-3$ is divisible by $x+4$. $k=$ [ANS]", "answer_v1": [ "275/16" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find $k$ such that the Polynomial $P(x)=-5x^3+k x^2-3x+3$ is divisible by $x-3$. $k=$ [ANS]", "answer_v2": [ "47/3" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find $k$ such that the Polynomial $P(x)=-4x^3+k x^2-4x-3$ is divisible by $x+5$. $k=$ [ANS]", "answer_v3": [ "-517/25" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0392", "subject": "Algebra", "topic": "Operations on polynomial and rational expressions", "subtopic": "Rational expressions: multiply, divide", "level": "2", "keywords": [ "rational", "expressions" ], "problem_v1": "Perform the indicated operation and simplify if possible. Assume all cancelled factors are not zero.\n$\\begin{array}{cccc}\\hline & \\frac{16}{p^2} \\cdot \\frac{4 p^2 q^2}{p^4}\\div \\frac{8 p^4}{p^6}=& & [ANS] [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "8*q^2", "p^2" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Perform the indicated operation and simplify if possible. Assume all cancelled factors are not zero.\n$\\begin{array}{cccc}\\hline & \\frac{24}{x} \\div \\frac{3x y}{x^2}\\cdot \\frac{12x^2}{x^4}=& & [ANS] [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "96", "x^2*y" ], "answer_type_v2": [ "NV", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Perform the indicated operation and simplify if possible. Assume all cancelled factors are not zero.\n$\\begin{array}{cccc}\\hline & \\frac{18}{s^2} \\div \\frac{3 s^2 t^2}{s^5}\\cdot \\frac{6 s^5}{s^8}=& & [ANS] [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "36", "s^2*t^2" ], "answer_type_v3": [ "NV", "EX" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0393", "subject": "Algebra", "topic": "Operations on polynomial and rational expressions", "subtopic": "Rational expressions: add, subtract", "level": "2", "keywords": [ "least common denominator" ], "problem_v1": "Perform the indicated operation, and simplify if possible. Write your answer without using negative exponents.\n$\\begin{array}{cccc}\\hline & (3 b)^{-2}+(4 b)^{-1}=& & [ANS] [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "9*b+4", "36*b^2" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Perform the indicated operation, and simplify if possible. Write your answer without using negative exponents.\n$\\begin{array}{cccc}\\hline & (5x)^{-1}+x^{-2}=& & [ANS] [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "x+5", "5*x^2" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Perform the indicated operation, and simplify if possible. Write your answer without using negative exponents.\n$\\begin{array}{cccc}\\hline & (4 y)^{-2}+(2 y)^{-1}=& & [ANS] [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "8*y+1", "16*y^2" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0394", "subject": "Algebra", "topic": "Operations on polynomial and rational expressions", "subtopic": "Simplify rational expressions", "level": "2", "keywords": [ "functions" ], "problem_v1": "Part 1 of 2:\nExpand and simplify $ B(v)=40 \\left(\\frac{v-16 v^2}{v} \\right)$.\n$B(v)$=[ANS]", "answer_v1": [ "40-640*v" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Part 1 of 2:\nExpand and simplify $ B(v)=20 \\left(\\frac{v-18 v^2}{v} \\right)$.\n$B(v)$=[ANS]", "answer_v2": [ "20-360*v" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Part 1 of 2:\nExpand and simplify $ B(v)=20 \\left(\\frac{v-17 v^2}{v} \\right)$.\n$B(v)$=[ANS]", "answer_v3": [ "20-340*v" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0395", "subject": "Algebra", "topic": "Operations on polynomial and rational expressions", "subtopic": "Simplify rational expressions", "level": "2", "keywords": [ "rational", "expressions" ], "problem_v1": "Consider the expression\n \\frac{-3+p}{p} List the letter(s) corresponding to the expressions that are equivalent to the one above. A. $ \\frac{p-3}{-p}$ B. $ \\frac{p-3}{p}$ C. $-\\frac{3-p}{p}$ D. $ \\frac{3-p}{-p}$ Answer(s): [ANS]", "answer_v1": [ "BCD" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "Consider the expression\n \\frac{x}{13-x} List the letter(s) corresponding to the expressions that are equivalent to the one above. A. $ \\frac{-x}{-13+x}$ B. $ \\frac{-x}{x-13}$ C. $ \\frac{-x}{13-x}$ D. $ \\frac{x}{x-13}$ Answer(s): [ANS]", "answer_v2": [ "AB" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "Consider the expression\n \\frac{s}{3-s} List the letter(s) corresponding to the expressions that are equivalent to the one above. A. $ \\frac{s}{s-3}$ B. $ \\frac{-s}{3-s}$ C. $ \\frac{-s}{-3+s}$ D. $ \\frac{-s}{s-3}$ Answer(s): [ANS]", "answer_v3": [ "CD" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Algebra_0396", "subject": "Algebra", "topic": "Operations on polynomial and rational expressions", "subtopic": "Partial fractions", "level": "4", "keywords": [ "algebra", "partial fractions" ], "problem_v1": "The partial fraction decomposition of $ \\frac{x^2+61}{x^3+x^2}$ can be written in the form of $ \\frac{f(x)}{x}+\\frac{g(x)}{x^2}+\\frac{h(x)}{x+1},$ where $f(x)=$ [ANS], $g(x)=$ [ANS], $h(x)=$ [ANS].", "answer_v1": [ "-61", "61", "62" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The partial fraction decomposition of $ \\frac{x^2+8}{x^3+x^2}$ can be written in the form of $ \\frac{f(x)}{x}+\\frac{g(x)}{x^2}+\\frac{h(x)}{x+1},$ where $f(x)=$ [ANS], $g(x)=$ [ANS], $h(x)=$ [ANS].", "answer_v2": [ "-8", "8", "9" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The partial fraction decomposition of $ \\frac{x^2+26}{x^3+x^2}$ can be written in the form of $ \\frac{f(x)}{x}+\\frac{g(x)}{x^2}+\\frac{h(x)}{x+1},$ where $f(x)=$ [ANS], $g(x)=$ [ANS], $h(x)=$ [ANS].", "answer_v3": [ "-26", "26", "27" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0397", "subject": "Algebra", "topic": "Polynomial equations and functions", "subtopic": "Polynomial equations", "level": "2", "keywords": [ "financial mathematics", "algebra" ], "problem_v1": "Find all values of $i$ that satisfy the equation 5=10 (1+i)^8. List the values below, separated by commas. (Note: you may need to carry your answers to several decimal places.) Values of $i$=[ANS]", "answer_v1": [ "(-0.0829959567953288, -1.91700404320467)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Find all values of $i$ that satisfy the equation 2=12 (1+i)^5. List the values below, separated by commas. (Note: you may need to carry your answers to several decimal places.) Values of $i$=[ANS]", "answer_v2": [ "-0.301172881228421" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find all values of $i$ that satisfy the equation 3=10 (1+i)^6. List the values below, separated by commas. (Note: you may need to carry your answers to several decimal places.) Values of $i$=[ANS]", "answer_v3": [ "(-0.181811176999973, -1.81818882300003)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Algebra_0398", "subject": "Algebra", "topic": "Polynomial equations and functions", "subtopic": "Polynomial equations", "level": "2", "keywords": [ "algebra", "polynomials", "division" ], "problem_v1": "Given that $x=1$ is a zero of the polynomial $P(x)=8x^3-8x^2-32x+32$, find all other zeros of the polynomial $P(x)$. The other zeros of $P(x)$ are $x=$ [ANS]. \u00a0 If there are multiple zeros, separate the different zeros by commas.", "answer_v1": [ "(2, -2)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Given that $x=5$ is a zero of the polynomial $P(x)=2x^3+2x^2-44x-80$, find all other zeros of the polynomial $P(x)$. The other zeros of $P(x)$ are $x=$ [ANS]. \u00a0 If there are multiple zeros, separate the different zeros by commas.", "answer_v2": [ "(-4, -2)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Given that $x=1$ is a zero of the polynomial $P(x)=4x^3+16x^2+4x-24$, find all other zeros of the polynomial $P(x)$. The other zeros of $P(x)$ are $x=$ [ANS]. \u00a0 If there are multiple zeros, separate the different zeros by commas.", "answer_v3": [ "(-2, -3)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Algebra_0399", "subject": "Algebra", "topic": "Polynomial equations and functions", "subtopic": "Polynomial equations", "level": "2", "keywords": [ "grouping", "factoring", "equations" ], "problem_v1": "Solve the equation for $a$:\n4 a^6=64 a^4 $a=$ [ANS]\nHelp: Separate multiple answers by a comma separated list.", "answer_v1": [ "(0, 4, -4)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Solve the equation for $x$:\n6x^4=24x^2 $x=$ [ANS]\nHelp: Separate multiple answers by a comma separated list.", "answer_v2": [ "(0, 2, -2)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Solve the equation for $z$:\n5 z^5=45 z^3 $z=$ [ANS]\nHelp: Separate multiple answers by a comma separated list.", "answer_v3": [ "(0, 3, -3)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Algebra_0400", "subject": "Algebra", "topic": "Polynomial equations and functions", "subtopic": "Polynomial functions", "level": "2", "keywords": [ "polynomial", "exponent", "binomial" ], "problem_v1": "Is the following expression a monomial, binomial, or trinomial?\n${3t^{15}+5t^{10}}$ [ANS] What is the degree of the expression? [ANS]", "answer_v1": [ "binomial", "15" ], "answer_type_v1": [ "MCS", "NV" ], "options_v1": [ [ "monomial", "binomial", "trinomial" ], [] ], "problem_v2": "Is the following expression a monomial, binomial, or trinomial?\n${18x^{7}-14x}$ [ANS] What is the degree of the expression? [ANS]", "answer_v2": [ "binomial", "7" ], "answer_type_v2": [ "MCS", "NV" ], "options_v2": [ [ "monomial", "binomial", "trinomial" ], [] ], "problem_v3": "Is the following expression a monomial, binomial, or trinomial?\n${4y^{11}-9y^{8}}$ [ANS] What is the degree of the expression? [ANS]", "answer_v3": [ "binomial", "11" ], "answer_type_v3": [ "MCS", "NV" ], "options_v3": [ [ "monomial", "binomial", "trinomial" ], [] ] }, { "id": "Algebra_0401", "subject": "Algebra", "topic": "Polynomial equations and functions", "subtopic": "Polynomial functions", "level": "2", "keywords": [ "polynomial", "zeros", "long-run behavior", "degree" ], "problem_v1": "Are the functions below polynomials? If they are, enter their degree. If not, enter NONE. $f(x)=x^{8}+4$ has degree [ANS]\n$g(x)=8^x+4$ has degree [ANS]", "answer_v1": [ "8", "NONE" ], "answer_type_v1": [ "NV", "OE" ], "options_v1": [ [], [] ], "problem_v2": "Are the functions below polynomials? If they are, enter their degree. If not, enter NONE. $f(x)=2^x+5$ has degree [ANS]\n$g(x)=x^{2}+5$ has degree [ANS]", "answer_v2": [ "NONE", "2" ], "answer_type_v2": [ "OE", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Are the functions below polynomials? If they are, enter their degree. If not, enter NONE. $f(x)=4^x+4$ has degree [ANS]\n$g(x)=x^{4}+4$ has degree [ANS]", "answer_v3": [ "NONE", "4" ], "answer_type_v3": [ "OE", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0402", "subject": "Algebra", "topic": "Polynomial equations and functions", "subtopic": "Polynomial functions", "level": "2", "keywords": [ "polynomials" ], "problem_v1": "Determine the following for: $5x^{4} y^{5}+x^{3} y^{7}+2$ a) Determine the term and coefficient of each degree.\n$\\begin{array}{ccc}\\hline Term & Coefficient & Degree \\\\ \\hline 5x^{4} y^{5} & [ANS] & [ANS] \\\\ \\hline x^{3} y^{7} & [ANS] & [ANS] \\\\ \\hline 2 & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nb) The degree of the polynomial is [ANS], the leading term is [ANS], and the leading coefficient is [ANS].", "answer_v1": [ "5", "9", "1", "10", "2", "0", "10", "1*x**3*y**7", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Determine the following for: $x^{4} y^{4}+x^{2} y^{10}-4$ a) Determine the term and coefficient of each degree.\n$\\begin{array}{ccc}\\hline Term & Coefficient & Degree \\\\ \\hline x^{4} y^{4} & [ANS] & [ANS] \\\\ \\hline x^{2} y^{10} & [ANS] & [ANS] \\\\ \\hline-4 & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nb) The degree of the polynomial is [ANS], the leading term is [ANS], and the leading coefficient is [ANS].", "answer_v2": [ "1", "8", "1", "12", "-4", "0", "12", "1*x**2*y**10", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Determine the following for: $x^{4} y^{4}+x^{3} y^{7}-4$ a) Determine the term and coefficient of each degree.\n$\\begin{array}{ccc}\\hline Term & Coefficient & Degree \\\\ \\hline x^{4} y^{4} & [ANS] & [ANS] \\\\ \\hline x^{3} y^{7} & [ANS] & [ANS] \\\\ \\hline-4 & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nb) The degree of the polynomial is [ANS], the leading term is [ANS], and the leading coefficient is [ANS].", "answer_v3": [ "1", "8", "1", "10", "-4", "0", "10", "1*x**3*y**7", "1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Algebra_0403", "subject": "Algebra", "topic": "Polynomial equations and functions", "subtopic": "Polynomial functions", "level": "1", "keywords": [ "algebra" ], "problem_v1": "Match the verbal descriptions with the given polynomials. You need to use all polynomials and all descriptions. Recall that polynomials of degrees 0, 1, 2, 3, 4, 5, are called constant, linear, quadratic, cubic, quartic, and quintic, respectively. Also recall the definitions of the terms monomial, binomial, trinomial, given. You must get all of the answers correct to receive credit. [ANS] 1. A cubic polynomial [ANS] 2. A trinomial [ANS] 3. A quartic binomial [ANS] 4. The square of a cubic polynomial [ANS] 5. A quintic monomial.\nA. $x^2+2x+1$ B. $(x^3+1)^2$ C. $x^4-2x^3$ D. $x^3+3x^2+3x+1$ E. $\\pi x^5$.", "answer_v1": [ "D", "A", "C", "B", "E" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Match the verbal descriptions with the given polynomials. You need to use all polynomials and all descriptions. Recall that polynomials of degrees 0, 1, 2, 3, 4, 5, are called constant, linear, quadratic, cubic, quartic, and quintic, respectively. Also recall the definitions of the terms monomial, binomial, trinomial, given. You must get all of the answers correct to receive credit. [ANS] 1. A quintic monomial. [ANS] 2. The square of a cubic polynomial [ANS] 3. A quartic binomial [ANS] 4. A trinomial [ANS] 5. A cubic polynomial\nA. $(x^3+1)^2$ B. $\\pi x^5$ C. $x^4-2x^3$ D. $x^3+3x^2+3x+1$ E. $x^2+2x+1$.", "answer_v2": [ "B", "A", "C", "E", "D" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Match the verbal descriptions with the given polynomials. You need to use all polynomials and all descriptions. Recall that polynomials of degrees 0, 1, 2, 3, 4, 5, are called constant, linear, quadratic, cubic, quartic, and quintic, respectively. Also recall the definitions of the terms monomial, binomial, trinomial, given. You must get all of the answers correct to receive credit. [ANS] 1. A quartic binomial [ANS] 2. A cubic polynomial [ANS] 3. A quintic monomial. [ANS] 4. The square of a cubic polynomial [ANS] 5. A trinomial\nA. $x^3+3x^2+3x+1$ B. $x^2+2x+1$ C. $(x^3+1)^2$ D. $\\pi x^5$ E. $x^4-2x^3$.", "answer_v3": [ "E", "A", "D", "C", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Algebra_0404", "subject": "Algebra", "topic": "Polynomial equations and functions", "subtopic": "Polynomial functions", "level": "", "keywords": [ "Bernstein polynomials", "degree" ], "problem_v1": "Write the formula for the 1st Bernstein polynomial of degree 3. For all parts, express your answer in the variable $t$. Answer: [ANS]\nWrite the formula for the 2nd Bernstein polynomial of degree 4. Answer: [ANS]\nWrite the formula for the 3rd Bernstein polynomial of degree 5. Answer: [ANS]\nWrite the formula for the 6th Bernstein polynomial of degree 7. Answer: [ANS]\nWrite the formula for the 4th Bernstein polynomial of degree 8. Answer: [ANS]", "answer_v1": [ "3*t^1*(1-t)^2", "6*t^2*(1-t)^2", "10*t^3*(1-t)^2", "7*t^6*(1-t)^1", "70*t^4*(1-t)^4" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Write the formula for the 0th Bernstein polynomial of degree 2. For all parts, express your answer in the variable $t$. Answer: [ANS]\nWrite the formula for the 1st Bernstein polynomial of degree 3. Answer: [ANS]\nWrite the formula for the 2nd Bernstein polynomial of degree 3. Answer: [ANS]\nWrite the formula for the 5th Bernstein polynomial of degree 6. Answer: [ANS]\nWrite the formula for the 4th Bernstein polynomial of degree 10. Answer: [ANS]", "answer_v2": [ "1*t^0*(1-t)^2", "3*t^1*(1-t)^2", "3*t^2*(1-t)^1", "6*t^5*(1-t)^1", "210*t^4*(1-t)^6" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Write the formula for the 1st Bernstein polynomial of degree 2. For all parts, express your answer in the variable $t$. Answer: [ANS]\nWrite the formula for the 2nd Bernstein polynomial of degree 3. Answer: [ANS]\nWrite the formula for the 0th Bernstein polynomial of degree 3. Answer: [ANS]\nWrite the formula for the 5th Bernstein polynomial of degree 6. Answer: [ANS]\nWrite the formula for the 4th Bernstein polynomial of degree 8. Answer: [ANS]", "answer_v3": [ "2*t^1*(1-t)^1", "3*t^2*(1-t)^1", "1*t^0*(1-t)^3", "6*t^5*(1-t)^1", "70*t^4*(1-t)^4" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Algebra_0405", "subject": "Algebra", "topic": "Polynomial equations and functions", "subtopic": "Polynomial functions", "level": "3", "keywords": [ "algebra", "horner", "nested", "factorization" ], "problem_v1": "Suppose\n$f(x)=-26-39x^1+37x^2+27x^3+10x^4+1x^5$\nis factored using Horner's Method Horner's Method into the nested form\n$f({\\color{Blue} x})=-26+{\\color{Blue} x}$ $(-39+{\\color{Blue} x}$ $(37+{\\color{Blue} x}$ $(27+{\\color{Blue} x}$ $(10+{\\color{Blue} x}$ $(1)$ $)$ $)$ $)$ $)$\nThen\n$f({\\color{Blue}-7})=-26+{\\color{Blue}-7}$ $(-39+{\\color{Blue}-7}$ $(37+{\\color{Blue}-7}$ $(27+{\\color{Blue}-7}$ $(10+{\\color{Blue}-7}$ $($ [ANS] $)$ $)$ $)$ $)$ $)$ $f({\\color{Blue}-7})=-26+{\\color{Blue}-7}$ $(-39+{\\color{Blue}-7}$ $(37+{\\color{Blue}-7}$ $(27+{\\color{Blue}-7}$ $($ [ANS] $)$ $)$ $)$ $)$ $f({\\color{Blue}-7})=-26+{\\color{Blue}-7}$ $(-39+{\\color{Blue}-7}$ $(37+{\\color{Blue}-7}$ $($ [ANS] $)$ $)$ $)$ $f({\\color{Blue}-7})=-26+{\\color{Blue}-7}$ $(-39+{\\color{Blue}-7}$ $($ [ANS] $)$ $)$ $f({\\color{Blue}-7})=-26+{\\color{Blue}-7}$ $($ [ANS] $)$ $f({\\color{Blue}-7})=$ [ANS]", "answer_v1": [ "1", "10+-7*1", "27+-7*3", "37+-7*6", "-39+-7*-5", "-26+-7*-4" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Suppose\n$f(x)=-18+17x^1+3x^2-22x^3-1x^4+4x^5$\nis factored using Horner's Method Horner's Method into the nested form\n$f({\\color{Blue} x})=-18+{\\color{Blue} x}$ $(17+{\\color{Blue} x}$ $(3+{\\color{Blue} x}$ $(-22+{\\color{Blue} x}$ $(-1+{\\color{Blue} x}$ $(4)$ $)$ $)$ $)$ $)$\nThen\n$f({\\color{Blue}-2})=-18+{\\color{Blue}-2}$ $(17+{\\color{Blue}-2}$ $(3+{\\color{Blue}-2}$ $(-22+{\\color{Blue}-2}$ $(-1+{\\color{Blue}-2}$ $($ [ANS] $)$ $)$ $)$ $)$ $)$ $f({\\color{Blue}-2})=-18+{\\color{Blue}-2}$ $(17+{\\color{Blue}-2}$ $(3+{\\color{Blue}-2}$ $(-22+{\\color{Blue}-2}$ $($ [ANS] $)$ $)$ $)$ $)$ $f({\\color{Blue}-2})=-18+{\\color{Blue}-2}$ $(17+{\\color{Blue}-2}$ $(3+{\\color{Blue}-2}$ $($ [ANS] $)$ $)$ $)$ $f({\\color{Blue}-2})=-18+{\\color{Blue}-2}$ $(17+{\\color{Blue}-2}$ $($ [ANS] $)$ $)$ $f({\\color{Blue}-2})=-18+{\\color{Blue}-2}$ $($ [ANS] $)$ $f({\\color{Blue}-2})=$ [ANS]", "answer_v2": [ "4", "-1+-2*4", "-22+-2*-9", "3+-2*-4", "17+-2*11", "-18+-2*-5" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Suppose\n$f(x)=-8-32x^1-3x^2-23x^3-2x^4+1x^5$\nis factored using Horner's Method Horner's Method into the nested form\n$f({\\color{Blue} x})=-8+{\\color{Blue} x}$ $(-32+{\\color{Blue} x}$ $(-3+{\\color{Blue} x}$ $(-23+{\\color{Blue} x}$ $(-2+{\\color{Blue} x}$ $(1)$ $)$ $)$ $)$ $)$\nThen\n$f({\\color{Blue}-4})=-8+{\\color{Blue}-4}$ $(-32+{\\color{Blue}-4}$ $(-3+{\\color{Blue}-4}$ $(-23+{\\color{Blue}-4}$ $(-2+{\\color{Blue}-4}$ $($ [ANS] $)$ $)$ $)$ $)$ $)$ $f({\\color{Blue}-4})=-8+{\\color{Blue}-4}$ $(-32+{\\color{Blue}-4}$ $(-3+{\\color{Blue}-4}$ $(-23+{\\color{Blue}-4}$ $($ [ANS] $)$ $)$ $)$ $)$ $f({\\color{Blue}-4})=-8+{\\color{Blue}-4}$ $(-32+{\\color{Blue}-4}$ $(-3+{\\color{Blue}-4}$ $($ [ANS] $)$ $)$ $)$ $f({\\color{Blue}-4})=-8+{\\color{Blue}-4}$ $(-32+{\\color{Blue}-4}$ $($ [ANS] $)$ $)$ $f({\\color{Blue}-4})=-8+{\\color{Blue}-4}$ $($ [ANS] $)$ $f({\\color{Blue}-4})=$ [ANS]", "answer_v3": [ "1", "-2+-4*1", "-23+-4*-6", "-3+-4*1", "-32+-4*-7", "-8+-4*-4" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Algebra_0406", "subject": "Algebra", "topic": "Polynomial equations and functions", "subtopic": "Remainder and factor theorems", "level": "3", "keywords": [ "logarithms", "exponentials", "exponential growth", "decay" ], "problem_v1": "List all possible rational roots for the function f(x)=7x^4+2x^3-4x^2+4x+119. Give your list in increasing order. Beside each possible rational root, type \"yes\" if it is a root and \"no\" if it is not a root. Leave any unnecessary answer blanks empty.\nPossible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS].", "answer_v1": [ "-119", "NO", "-17", "no", "-7", "no", "-2.42857142857143", "no", "-1", "no", "-0.142857142857143", "no", "0.142857142857143", "no", "1", "no", "2.42857142857143", "no", "7", "no", "17", "no", "119", "no" ], "answer_type_v1": [ "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "List all possible rational roots for the function f(x)=2x^4-7x^3+8x^2-3x+38. Give your list in increasing order. Beside each possible rational root, type \"yes\" if it is a root and \"no\" if it is not a root. Leave any unnecessary answer blanks empty.\nPossible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS].", "answer_v2": [ "-38", "NO", "-19", "no", "-9.5", "no", "-2", "no", "-1", "no", "-0.5", "no", "0.5", "no", "1", "no", "2", "no", "9.5", "no", "19", "no", "38", "no" ], "answer_type_v2": [ "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "List all possible rational roots for the function f(x)=3x^4-4x^3-6x^2+1x+51. Give your list in increasing order. Beside each possible rational root, type \"yes\" if it is a root and \"no\" if it is not a root. Leave any unnecessary answer blanks empty.\nPossible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS]. Possible rational root: [ANS] Is it a root? [ANS].", "answer_v3": [ "-51", "NO", "-17", "no", "-5.66666666666667", "no", "-3", "no", "-1", "no", "-0.333333333333333", "no", "0.333333333333333", "no", "1", "no", "3", "no", "5.66666666666667", "no", "17", "no", "51", "no" ], "answer_type_v3": [ "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Algebra_0407", "subject": "Algebra", "topic": "Polynomial equations and functions", "subtopic": "Counting zeros", "level": "2", "keywords": [ "logarithms", "exponentials", "exponential growth", "decay" ], "problem_v1": "Given the following table of values, what is the minimum number of roots that $f(x)$ can have?\n$\\begin{array}{cccccc}\\hline x &-4 &-2 & 0 & 2 & 4 \\\\ \\hline f(x) & 5 & 2 & 2 & 4 &-4 \\\\ \\hline \\end{array}$\n$f(x)$ has at least [ANS] root(s).", "answer_v1": [ "1" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Given the following table of values, what is the minimum number of roots that $f(x)$ can have?\n$\\begin{array}{cccccc}\\hline x &-4 &-2 & 0 & 2 & 4 \\\\ \\hline f(x) &-8 & 8 &-7 &-3 & 8 \\\\ \\hline \\end{array}$\n$f(x)$ has at least [ANS] root(s).", "answer_v2": [ "3" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Given the following table of values, what is the minimum number of roots that $f(x)$ can have?\n$\\begin{array}{cccccc}\\hline x &-4 &-2 & 0 & 2 & 4 \\\\ \\hline f(x) &-4 & 2 &-4 & 1 &-6 \\\\ \\hline \\end{array}$\n$f(x)$ has at least [ANS] root(s).", "answer_v3": [ "4" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0408", "subject": "Algebra", "topic": "Polynomial equations and functions", "subtopic": "Counting zeros", "level": "3", "keywords": [ "logarithms", "exponentials", "exponential growth", "decay" ], "problem_v1": "Let p(x)=x^{6}-14x^{5}+72x^{4}-78x^{3}-933x^{2}+5372x-9860. The polynomial $p(x)$ has exactly one positive real root. Between what two consecutive integers does it lie?\nThe positive root is between [ANS] and [ANS].", "answer_v1": [ "4", "5" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let p(x)=x^{6}+30x^{5}+349x^{4}+890x^{3}-17586x^{2}-165920x-452864. The polynomial $p(x)$ has exactly one positive real root. Between what two consecutive integers does it lie?\nThe positive root is between [ANS] and [ANS].", "answer_v2": [ "7", "8" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let p(x)=x^{6}+16x^{5}+90x^{4}+120x^{3}-771x^{2}-3256x-3740. The polynomial $p(x)$ has exactly one positive real root. Between what two consecutive integers does it lie?\nThe positive root is between [ANS] and [ANS].", "answer_v3": [ "3", "4" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0409", "subject": "Algebra", "topic": "Polynomial equations and functions", "subtopic": "Graphs of polynomials", "level": "3", "keywords": [ "polynomial", "zeros", "long-run behavior", "degree" ], "problem_v1": "Which of the formulas below could be a polynomial with all of the following properties: its only zeros are $x=-6,-2, 4$, it has $y$-intercept $y=6$, and its long-run behavior is $y \\to-\\infty$ as $x \\to \\pm \\infty$? Select every formula that has all of these properties. [ANS] A. $y=-\\frac{6}{192} (x+6)(x+2)(x-4)^2$ B. $y=-\\frac{6}{96} (x+6)(x+2)^2(x-4)$ C. $y=-6x(x+6)(x+2)(x-4)$ D. $y=-\\frac{6}{48} (x+6)(x+2)(x-4)$ E. $y=-\\frac{6}{288} (x+6)^2(x+2)(x-4)$ F. $y=-\\frac{6}{10368} (x+6)^4(x+2)(x-4)$ G. $y=-\\frac{6}{288} (x-6)(x-2)(x+4)$", "answer_v1": [ "BEF" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v2": "Which of the formulas below could be a polynomial with all of the following properties: its only zeros are $x=-9,-1, 2$, it has $y$-intercept $y=4$, and its long-run behavior is $y \\to-\\infty$ as $x \\to \\pm \\infty$? Select every formula that has all of these properties. [ANS] A. $y=-\\frac{4}{162} (x-9)(x-1)(x+2)$ B. $y=-\\frac{4}{18} (x+9)(x+1)^2(x-2)$ C. $y=-4x(x+9)(x+1)(x-2)$ D. $y=-\\frac{4}{162} (x+9)^2(x+1)(x-2)$ E. $y=-\\frac{4}{18} (x+9)(x+1)^4(x-2)$ F. $y=\\frac{4}{36} (x+9)(x+1)(x-2)^2$ G. $y=-\\frac{4}{36} (x+9)(x+1)(x-2)^2$", "answer_v2": [ "BDE" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v3": "Which of the formulas below could be a polynomial with all of the following properties: its only zeros are $x=-8,-2, 3$, it has $y$-intercept $y=5$, and its long-run behavior is $y \\to-\\infty$ as $x \\to \\pm \\infty$? Select every formula that has all of these properties. [ANS] A. $y=-\\frac{5}{48} (x+8)(x+2)(x-3)$ B. $y=-5x(x+8)(x+2)(x-3)$ C. $y=\\frac{5}{144} (x+8)(x+2)(x-3)^2$ D. $y=-\\frac{5}{384} (x+8)^2(x+2)(x-3)$ E. $y=-\\frac{5}{96} (x+8)(x+2)^2(x-3)$ F. $y=-\\frac{5}{384} (x-8)(x-2)(x+3)$ G. $y=-\\frac{5}{24576} (x+8)^4(x+2)(x-3)$", "answer_v3": [ "DEG" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G" ] ] }, { "id": "Algebra_0410", "subject": "Algebra", "topic": "Polynomial equations and functions", "subtopic": "Graphs of polynomials", "level": "3", "keywords": [ "algebra", "functions", "graphing calculator" ], "problem_v1": "Use a graphing calculator to approximate to two decimal places, the real solutions to the equation.\nx^4-0.27x^3+6.676x^2-1.89x-2.268=0. $x$=[ANS]\nNote: If there is more than one solution write them separated by commas.", "answer_v1": [ "(0.72, -0.45)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Use a graphing calculator to approximate to two decimal places, the real solutions to the equation.\nx^4-0.03x^3+1.9622x^2-0.06x-0.0756=0. $x$=[ANS]\nNote: If there is more than one solution write them separated by commas.", "answer_v2": [ "(0.21, -0.18)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Use a graphing calculator to approximate to two decimal places, the real solutions to the equation.\nx^4+0.06x^3+2.8245x^2+0.18x-0.5265=0. $x$=[ANS]\nNote: If there is more than one solution write them separated by commas.", "answer_v3": [ "(0.39, -0.45)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Algebra_0411", "subject": "Algebra", "topic": "Polynomial equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "polynomial", "add", "simplify", "binomial", "combine", "like terms" ], "problem_v1": "An architect is designing a house on an empty plot. The area of the plot can be modeled by the polynomial ${5x^{4}+14x^{2}+2.5x}$, and the area of the house\u2019s base can be modeled by ${3x^{3}+2.5x-20}$. The rest of the plot is the yard. What\u2019s the yard\u2019s area? The area of the yard can be modeled by the polynomial [ANS].", "answer_v1": [ "5*x^4-3*x^3+14*x^2+20" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "An architect is designing a house on an empty plot. The area of the plot can be modeled by the polynomial ${2x^{4}+19x^{2}-7x}$, and the area of the house\u2019s base can be modeled by ${3x^{3}-7x+45}$. The rest of the plot is the yard. What\u2019s the yard\u2019s area? The area of the yard can be modeled by the polynomial [ANS].", "answer_v2": [ "2*x^4-3*x^3+19*x^2-45" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "An architect is designing a house on an empty plot. The area of the plot can be modeled by the polynomial ${3x^{4}+14x^{2}-4.5x}$, and the area of the house\u2019s base can be modeled by ${4x^{3}-4.5x-30}$. The rest of the plot is the yard. What\u2019s the yard\u2019s area? The area of the yard can be modeled by the polynomial [ANS].", "answer_v3": [ "3*x^4-4*x^3+14*x^2+30" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0412", "subject": "Algebra", "topic": "Polynomial equations and functions", "subtopic": "Applications and models", "level": "2", "keywords": [ "polynomial", "zeros", "long-run behavior", "degree" ], "problem_v1": "The town of Smallsville was founded in 1900. Its population $y$ (in hundreds) is given by the equation y=-0.1x^4+1.7x^3-9x^2+14.4x+6, where $x$ is the number of years since 1900. Use a the graph in the window $0 \\le x \\le 10$, $-2 \\le y \\le 15$. a) What was the population of Smallsville when it was founded? [ANS] people b) When did Smallsville become a ghost town (nobody lived there anymore)? In [ANS] (month, do not abbreviate) of [ANS] (year) c) What was the largest population of Smallsville after 1905? [ANS] (round to nearest whole person) d) In what month did Smallsville reach that population? In [ANS] (month, do not abbreviate) of [ANS] (year)", "answer_v1": [ "600", "July", "1908", "890", "March", "1907" ], "answer_type_v1": [ "NV", "MCS", "NV", "NV", "MCS", "NV" ], "options_v1": [ [], [ "January", "February", "March", "April", "May", "June", "July", "August", "September", "October", "November", "December" ], [], [], [ "January", "February", "March", "April", "May", "June", "July", "August", "September", "October", "November", "December" ], [] ], "problem_v2": "The town of Smallsville was founded in 1900. Its population $y$ (in hundreds) is given by the equation y=-0.1x^4+0.5x^3+0.9x^2-4.5x+6, where $x$ is the number of years since 1900. Use a the graph in the window $0 \\le x \\le 10$, $-2 \\le y \\le 15$. a) What was the population of Smallsville when it was founded? [ANS] people b) When did Smallsville become a ghost town (nobody lived there anymore)? In [ANS] (month, do not abbreviate) of [ANS] (year) c) What was the largest population of Smallsville after 1903? [ANS] (round to nearest whole person) d) In what month did Smallsville reach that population? In [ANS] (month, do not abbreviate) of [ANS] (year)", "answer_v2": [ "600", "July", "1905", "890", "March", "1904" ], "answer_type_v2": [ "NV", "MCS", "NV", "NV", "MCS", "NV" ], "options_v2": [ [], [ "January", "February", "March", "April", "May", "June", "July", "August", "September", "October", "November", "December" ], [], [], [ "January", "February", "March", "April", "May", "June", "July", "August", "September", "October", "November", "December" ], [] ], "problem_v3": "The town of Smallsville was founded in 1900. Its population $y$ (in hundreds) is given by the equation y=-0.1x^4+0.9x^3-1.2x^2-4.4x+10.8, where $x$ is the number of years since 1900. Use a the graph in the window $0 \\le x \\le 10$, $-2 \\le y \\le 15$. a) What was the population of Smallsville when it was founded? [ANS] people b) When did Smallsville become a ghost town (nobody lived there anymore)? In [ANS] (month, do not abbreviate) of [ANS] (year) c) What was the largest population of Smallsville after 1904? [ANS] (round to nearest whole person) d) In what month did Smallsville reach that population? In [ANS] (month, do not abbreviate) of [ANS] (year)", "answer_v3": [ "1080", "July", "1906", "890", "March", "1905" ], "answer_type_v3": [ "NV", "MCS", "NV", "NV", "MCS", "NV" ], "options_v3": [ [], [ "January", "February", "March", "April", "May", "June", "July", "August", "September", "October", "November", "December" ], [], [], [ "January", "February", "March", "April", "May", "June", "July", "August", "September", "October", "November", "December" ], [] ] }, { "id": "Algebra_0413", "subject": "Algebra", "topic": "Polynomial equations and functions", "subtopic": "Applications and models", "level": "2", "keywords": [ "Sphere", "Volume", "Surface Area" ], "problem_v1": "As Aragorn views the Dark Lord in a crystal ball of radius 8, he realizes that: The surface area of the ball equals: [ANS]\nThe volume of the ball equals: [ANS]", "answer_v1": [ "804.247719318987", "2144.66058485063" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "As Aragorn views the Dark Lord in a crystal ball of radius 1, he realizes that: The surface area of the ball equals: [ANS]\nThe volume of the ball equals: [ANS]", "answer_v2": [ "12.5663706143592", "4.18879020478639" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "As Aragorn views the Dark Lord in a crystal ball of radius 4, he realizes that: The surface area of the ball equals: [ANS]\nThe volume of the ball equals: [ANS]", "answer_v3": [ "201.061929829747", "268.082573106329" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0414", "subject": "Algebra", "topic": "Polynomial equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "algebra", "polynomial function", "real zero", "story question" ], "problem_v1": "A grain silo consists of a cylindrical main section and a hemispherical roof. If the total volume of the silo (including the part inside the roof section) is 18000 $\\textrm{ft}^3$ and the cylindrical part is 40 ft tall, what is the radius of the silo?\nNote: The following formulas may be useful:\n\\mbox{Volume of a Cylinder}=\\pi r^2 h \\mbox{Volume of a Sphere}=\\frac{4}{3} \\pi r^3 Radius $=$ [ANS] ft", "answer_v1": [ "11.0020003296683" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A grain silo consists of a cylindrical main section and a hemispherical roof. If the total volume of the silo (including the part inside the roof section) is 10000 $\\textrm{ft}^3$ and the cylindrical part is 50 ft tall, what is the radius of the silo?\nNote: The following formulas may be useful:\n\\mbox{Volume of a Cylinder}=\\pi r^2 h \\mbox{Volume of a Sphere}=\\frac{4}{3} \\pi r^3 Radius $=$ [ANS] ft", "answer_v2": [ "7.60279569348485" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A grain silo consists of a cylindrical main section and a hemispherical roof. If the total volume of the silo (including the part inside the roof section) is 13000 $\\textrm{ft}^3$ and the cylindrical part is 40 ft tall, what is the radius of the silo?\nNote: The following formulas may be useful:\n\\mbox{Volume of a Cylinder}=\\pi r^2 h \\mbox{Volume of a Sphere}=\\frac{4}{3} \\pi r^3 Radius $=$ [ANS] ft", "answer_v3": [ "9.45355447082057" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0415", "subject": "Algebra", "topic": "Polynomial equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "polynomials" ], "problem_v1": "The polynomial function f(x)=-2212x^2+57,575x+107,896 models the cumulative number of deaths from AIDS in the United States, $f(x)$, where $x$ is the number of years after 1990.\na) Find $f(10)$=[ANS]\nb) In what year is the cumulative number of AIDS deaths in the United States equal to your answer in part a)? Answer: [ANS]", "answer_v1": [ "462446", "2000" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The polynomial function f(x)=-2212x^2+57,575x+107,896 models the cumulative number of deaths from AIDS in the United States, $f(x)$, where $x$ is the number of years after 1990.\na) Find $f(3)$=[ANS]\nb) In what year is the cumulative number of AIDS deaths in the United States equal to your answer in part a)? Answer: [ANS]", "answer_v2": [ "260713", "1993" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The polynomial function f(x)=-2212x^2+57,575x+107,896 models the cumulative number of deaths from AIDS in the United States, $f(x)$, where $x$ is the number of years after 1990.\na) Find $f(6)$=[ANS]\nb) In what year is the cumulative number of AIDS deaths in the United States equal to your answer in part a)? Answer: [ANS]", "answer_v3": [ "373714", "1996" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0416", "subject": "Algebra", "topic": "Polynomial equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "polynomials" ], "problem_v1": "From 1995 to 2002, the total investment capital for a certain natural gas company can be modelled by C(t)=1.753t^{3}-10.16t^{2}+59.24t+460.9 where $C(t)$ is the total investment capital (in millions of dollars) and $t$ is the number of years after 1995. a) Find $C(3)$=[ANS]\nb) In what year is the total investment capital equal to your answer, in millions of dollars, in part a)? Answer: [ANS]", "answer_v1": [ "594.511", "1998" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "From 1995 to 2002, the total investment capital for a certain natural gas company can be modelled by C(t)=1.084t^{3}-10.87t^{2}+58.3t+455 where $C(t)$ is the total investment capital (in millions of dollars) and $t$ is the number of years after 1995. a) Find $C(7)$=[ANS]\nb) In what year is the total investment capital equal to your answer, in millions of dollars, in part a)? Answer: [ANS]", "answer_v2": [ "702.282", "2002" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "From 1995 to 2002, the total investment capital for a certain natural gas company can be modelled by C(t)=1.314t^{3}-10.21t^{2}+58.56t+458.2 where $C(t)$ is the total investment capital (in millions of dollars) and $t$ is the number of years after 1995. a) Find $C(2)$=[ANS]\nb) In what year is the total investment capital equal to your answer, in millions of dollars, in part a)? Answer: [ANS]", "answer_v3": [ "544.992", "1997" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0417", "subject": "Algebra", "topic": "Polynomial equations and functions", "subtopic": "Applications and models", "level": "3", "keywords": [ "polynomial equations" ], "problem_v1": "A tree is supported by a wire anchored in the ground 13 feet from its base. The wire is 4 feet longer than the height that it reaches on the tree. How long is the wire? a) Use the Pythagorean theorem to write a polynomial equation to solve to answer the question. Use the variable $x$ to represent the length of the wire. Answer: [ANS]\nb) Solve the equation and give the length of the wire. Answer: [ANS] feet", "answer_v1": [ "13^2 + (x-4)^2 = x^2", "23.125" ], "answer_type_v1": [ "EQ", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A tree is supported by a wire anchored in the ground 5 feet from its base. The wire is 5 feet longer than the height that it reaches on the tree. How long is the wire? a) Use the Pythagorean theorem to write a polynomial equation to solve to answer the question. Use the variable $x$ to represent the length of the wire. Answer: [ANS]\nb) Solve the equation and give the length of the wire. Answer: [ANS] feet", "answer_v2": [ "5^2 + (x-5)^2 = x^2", "5" ], "answer_type_v2": [ "EQ", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A tree is supported by a wire anchored in the ground 8 feet from its base. The wire is 4 feet longer than the height that it reaches on the tree. How long is the wire? a) Use the Pythagorean theorem to write a polynomial equation to solve to answer the question. Use the variable $x$ to represent the length of the wire. Answer: [ANS]\nb) Solve the equation and give the length of the wire. Answer: [ANS] feet", "answer_v3": [ "8^2 + (x-4)^2 = x^2", "10" ], "answer_type_v3": [ "EQ", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0418", "subject": "Algebra", "topic": "Polynomial equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "algebra", "nonlinear inequality" ], "problem_v1": "The length of a cube was measured and found to be 28 cm with a possible error in measurement of at most 0.006 cm. What is the maximum error in using this value of the length to compute the volume of the cube.\nYour answer is that the maximum error must be less than [ANS].", "answer_v1": [ "14.115024216" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The length of a cube was measured and found to be 20 cm with a possible error in measurement of at most 0.008 cm. What is the maximum error in using this value of the length to compute the volume of the cube.\nYour answer is that the maximum error must be less than [ANS].", "answer_v2": [ "9.603840512" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The length of a cube was measured and found to be 23 cm with a possible error in measurement of at most 0.006 cm. What is the maximum error in using this value of the length to compute the volume of the cube.\nYour answer is that the maximum error must be less than [ANS].", "answer_v3": [ "9.524484216" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0419", "subject": "Algebra", "topic": "Polynomial equations and functions", "subtopic": "Applications and models", "level": "2", "keywords": [ "Algebra", "Functions", "Polynomial" ], "problem_v1": "A market analyst finds that if a company produces and sells $x$ mixers annually, the total profit in dollars is P(x)=9x+0.3x^2-0.0015x^3-427 Graph the function $P$ in an appropriate viewing rectangle and use the graph to answer the following.\nWhen just a few mixers are produced, the company loses money (i.e., profit is negative). For example $P(10)=-308.5$, so the company loses \\$308.50 if it produces and sells only 10 mixers. How many mixers must the company produce to break even?\nNumber of mixers=[ANS]\nDoes the profit increase indefinitely as more mixers are produced and sold, or is there a largest possible profit the firm could earn? If there is a maximum profit, enter that value. If profit could increase indefinitely, enter None.\nMaximum profit=\\$ [ANS]", "answer_v1": [ "27", "2613.75631729995" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A market analyst finds that if a company produces and sells $x$ garlic presses annually, the total profit in dollars is P(x)=5x+0.4x^2-0.0012x^3-326 Graph the function $P$ in an appropriate viewing rectangle and use the graph to answer the following.\nWhen just a few garlic presses are produced, the company loses money (i.e., profit is negative). For example $P(10)=-237.2$, so the company loses \\$237.20 if it produces and sells only 10 garlic presses. How many garlic presses must the company produce to break even?\nNumber of garlic presses=[ANS]\nDoes the profit increase indefinitely as more garlic presses are produced and sold, or is there a largest possible profit the firm could earn? If there is a maximum profit, enter that value. If profit could increase indefinitely, enter None.\nMaximum profit=\\$ [ANS]", "answer_v2": [ "24", "7384.71509461511" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A market analyst finds that if a company produces and sells $x$ blenders annually, the total profit in dollars is P(x)=6x+0.3x^2-0.0013x^3-324 Graph the function $P$ in an appropriate viewing rectangle and use the graph to answer the following.\nWhen just a few blenders are produced, the company loses money (i.e., profit is negative). For example $P(10)=-235.3$, so the company loses \\$235.30 if it produces and sells only 10 blenders. How many blenders must the company produce to break even?\nNumber of blenders=[ANS]\nDoes the profit increase indefinitely as more blenders are produced and sold, or is there a largest possible profit the firm could earn? If there is a maximum profit, enter that value. If profit could increase indefinitely, enter None.\nMaximum profit=\\$ [ANS]", "answer_v3": [ "26", "2994.66302260388" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0420", "subject": "Algebra", "topic": "Polynomial equations and functions", "subtopic": "Complex roots", "level": "3", "keywords": [ "algebra", "complex number" ], "problem_v1": "Give a polynomial with real coefficients of smallest degree which has roots at $1$, $1+2i$, and $-2-2i$, and which passes through the point $(0,51)$.\n$f(x)=$ [ANS].", "answer_v1": [ "(x-1)*(x^2-2*x+1+4)*[x^2-(-4)*x+4+4]*(-1.275)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Give a polynomial with real coefficients of smallest degree which has roots at $5$, $-4-2i$, and $5-2i$, and which passes through the point $(0,-84)$.\n$f(x)=$ [ANS].", "answer_v2": [ "(x-5)*[x^2-(-8)*x+16+4]*(x^2-10*x+25+4)*0.0289655" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Give a polynomial with real coefficients of smallest degree which has roots at $1$, $-2+i$, and $-3-2i$, and which passes through the point $(0,-37)$.\n$f(x)=$ [ANS].", "answer_v3": [ "(x-1)*[x^2-(-4)*x+4+1]*[x^2-(-6)*x+9+4]*0.569231" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0421", "subject": "Algebra", "topic": "Variation and power functions", "subtopic": "Direct variation", "level": "2", "keywords": [ "Equation", "Variation" ], "problem_v1": "Suppose $r$ varies directly with $t$ and that $r=30$ when $t=5$. What is the value of $r$ when $t=10$? $r=$ [ANS]", "answer_v1": [ "60" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose $r$ varies directly with $t$ and that $r=6$ when $t=2$. What is the value of $r$ when $t=12$? $r=$ [ANS]", "answer_v2": [ "36" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose $r$ varies directly with $t$ and that $r=12$ when $t=3$. What is the value of $r$ when $t=10$? $r=$ [ANS]", "answer_v3": [ "40" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0422", "subject": "Algebra", "topic": "Variation and power functions", "subtopic": "Direct variation", "level": "2", "keywords": [], "problem_v1": "Suppose $y$ varies directly as $x$ and $y=36$ when $x=9$. Which is an equation relating $x$ and $y$ [ANS] A. $y=-\\frac{1}{4} x$ B. $y=\\frac{1}{4} x$ C. $y=4x$ D. $y=-4x$", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "Suppose $y$ varies directly as $x$ and $y=30$ when $x=6$. Which is an equation relating $x$ and $y$ [ANS] A. $y=-\\frac{1}{5} x$ B. $y=\\frac{1}{5} x$ C. $y=-5x$ D. $y=5x$", "answer_v2": [ "D" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "Suppose $y$ varies directly as $x$ and $y=28$ when $x=7$. Which is an equation relating $x$ and $y$ [ANS] A. $y=-\\frac{1}{4} x$ B. $y=4x$ C. $y=-4x$ D. $y=\\frac{1}{4} x$", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Algebra_0423", "subject": "Algebra", "topic": "Variation and power functions", "subtopic": "Direct variation", "level": "2", "keywords": [ "functions" ], "problem_v1": "The length $m$, in inches, of a model train is directly proportional to the length $r$, in inches, of the corresponding real train.\n(a) Write an equation that expresses $m$ as a function of $r$, using $k$ for the constant of proportionality. [ANS]\n(b) A Z scale model train is 1/220th the size of a real train. What is the constant of proportionality? [ANS]\nWhat is the length, in feet, of a real locomotive if its Z scale model is $10$ inches long? [ANS] feet.\n(c) A G scale model train is 1/24th the size of a real train. What is the constant of proportionality? [ANS]\nWhat is the length, in inches, of a G scale model if its real locomotive is $75$ feet long? [ANS] inches.", "answer_v1": [ "m = k*r", "0.00454545", "183.333", "0.0416667", "37.5" ], "answer_type_v1": [ "EX", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "The length $m$, in inches, of a model train is directly proportional to the length $r$, in inches, of the corresponding real train.\n(a) Write an equation that expresses $m$ as a function of $r$, using $k$ for the constant of proportionality. [ANS]\n(b) An HO scale model train is 1/87th the size of a real train. What is the constant of proportionality? [ANS]\nWhat is the length, in feet, of a real locomotive if its HO scale model is $12$ inches long? [ANS] feet.\n(c) A Z scale model train is 1/220th the size of a real train. What is the constant of proportionality? [ANS]\nWhat is the length, in inches, of a Z scale model if its real locomotive is $60$ feet long? [ANS] inches.", "answer_v2": [ "m = k*r", "0.0114943", "87", "0.00454545", "3.27273" ], "answer_type_v2": [ "EX", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "The length $m$, in inches, of a model train is directly proportional to the length $r$, in inches, of the corresponding real train.\n(a) Write an equation that expresses $m$ as a function of $r$, using $k$ for the constant of proportionality. [ANS]\n(b) An N scale model train is 1/160th the size of a real train. What is the constant of proportionality? [ANS]\nWhat is the length, in feet, of a real locomotive if its N scale model is $11$ inches long? [ANS] feet.\n(c) An HO scale model train is 1/87th the size of a real train. What is the constant of proportionality? [ANS]\nWhat is the length, in inches, of an HO scale model if its real locomotive is $65$ feet long? [ANS] inches.", "answer_v3": [ "m = k*r", "0.00625", "146.667", "0.0114943", "8.96552" ], "answer_type_v3": [ "EX", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Algebra_0424", "subject": "Algebra", "topic": "Variation and power functions", "subtopic": "Direct variation", "level": "1", "keywords": [ "functions" ], "problem_v1": "Suppose $y=\\sqrt{7} \\cdot x$. Is $y$ directly proportional to $x$? If it is, enter the constant of proportionality. Otherwise, enter NO. NO. [ANS]", "answer_v1": [ "2.64575" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose $y=\\sqrt{2} \\cdot x$. Is $y$ directly proportional to $x$? If it is, enter the constant of proportionality. Otherwise, enter NO. NO. [ANS]", "answer_v2": [ "1.41421" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose $y=\\sqrt{3} \\cdot x$. Is $y$ directly proportional to $x$? If it is, enter the constant of proportionality. Otherwise, enter NO. NO. [ANS]", "answer_v3": [ "1.73205" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0425", "subject": "Algebra", "topic": "Variation and power functions", "subtopic": "Direct variation", "level": "3", "keywords": [ "functions" ], "problem_v1": "The required cooling capacity, in BTUs, for a room air conditioner is directly proportional to the area of the room being cooled. A room of $280$ square feet requires an air conditioner whose cooling capacity is $6440$ BTUs.\n(a) What is the constant of proportionality, and what are its units? [ANS] [ANS]\n(b) If an air conditioner has a cooling capacity of $9430$ BTUs, how large a room can it cool? [ANS] [ANS]", "answer_v1": [ "23", "BTUs per square foot", "410", "square feet" ], "answer_type_v1": [ "NV", "MCS", "NV", "MCS" ], "options_v1": [ [], [ "BTUs", "feet", "square feet", "BTUs per foot", "BTUs per square foot" ], [], [ "BTUs", "feet", "square feet", "BTUs per foot", "BTUs per square foot" ] ], "problem_v2": "The required cooling capacity, in BTUs, for a room air conditioner is directly proportional to the area of the room being cooled. A room of $300$ square feet requires an air conditioner whose cooling capacity is $4500$ BTUs.\n(a) What is the constant of proportionality, and what are its units? [ANS] [ANS]\n(b) If an air conditioner has a cooling capacity of $6000$ BTUs, how large a room can it cool? [ANS] [ANS]", "answer_v2": [ "15", "BTUs per square foot", "400", "square feet" ], "answer_type_v2": [ "NV", "MCS", "NV", "MCS" ], "options_v2": [ [], [ "BTUs", "feet", "square feet", "BTUs per foot", "BTUs per square foot" ], [], [ "BTUs", "feet", "square feet", "BTUs per foot", "BTUs per square foot" ] ], "problem_v3": "The required cooling capacity, in BTUs, for a room air conditioner is directly proportional to the area of the room being cooled. A room of $280$ square feet requires an air conditioner whose cooling capacity is $5040$ BTUs.\n(a) What is the constant of proportionality, and what are its units? [ANS] [ANS]\n(b) If an air conditioner has a cooling capacity of $7020$ BTUs, how large a room can it cool? [ANS] [ANS]", "answer_v3": [ "18", "BTUs per square foot", "390", "square feet" ], "answer_type_v3": [ "NV", "MCS", "NV", "MCS" ], "options_v3": [ [], [ "BTUs", "feet", "square feet", "BTUs per foot", "BTUs per square foot" ], [], [ "BTUs", "feet", "square feet", "BTUs per foot", "BTUs per square foot" ] ] }, { "id": "Algebra_0426", "subject": "Algebra", "topic": "Variation and power functions", "subtopic": "Direct variation", "level": "3", "keywords": [ "power functions" ], "problem_v1": "Suppose $A$ is directly proportional to the square of $B$, and $A=96$ when $B=4$.\nFind the constant of proportionality. [ANS]\nWrite the formula for $A$ in terms of $B$. [ANS]\nUse your formula to find $A$ when $B=7$. $A=$ [ANS]", "answer_v1": [ "6", "A = 6*B^2", "294" ], "answer_type_v1": [ "NV", "EQ", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Suppose $A$ is directly proportional to the square of $B$, and $A=75$ when $B=5$.\nFind the constant of proportionality. [ANS]\nWrite the formula for $A$ in terms of $B$. [ANS]\nUse your formula to find $A$ when $B=6$. $A=$ [ANS]", "answer_v2": [ "3", "A = 3*B^2", "108" ], "answer_type_v2": [ "NV", "EQ", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Suppose $A$ is directly proportional to the square of $B$, and $A=64$ when $B=4$.\nFind the constant of proportionality. [ANS]\nWrite the formula for $A$ in terms of $B$. [ANS]\nUse your formula to find $A$ when $B=6$. $A=$ [ANS]", "answer_v3": [ "4", "A = 4*B^2", "144" ], "answer_type_v3": [ "NV", "EQ", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0427", "subject": "Algebra", "topic": "Variation and power functions", "subtopic": "Direct variation", "level": "5", "keywords": [ "power functions" ], "problem_v1": "Identify the exponent and the coefficient in the following power function: The perimeter of a regular octagon of side $x$ is $P=8x$.\nExponent=[ANS]\nCoefficient=[ANS]", "answer_v1": [ "1", "8" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Identify the exponent and the coefficient in the following power function: The perimeter of a square of side $x$ is $P=4x$.\nExponent=[ANS]\nCoefficient=[ANS]", "answer_v2": [ "1", "4" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Identify the exponent and the coefficient in the following power function: The perimeter of a regular pentagon of side $x$ is $P=5x$.\nExponent=[ANS]\nCoefficient=[ANS]", "answer_v3": [ "1", "5" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0428", "subject": "Algebra", "topic": "Variation and power functions", "subtopic": "Inverse variation", "level": "2", "keywords": [ "Equation", "Variation" ], "problem_v1": "A company has found that the demand for its product varies inversely as the price of the product. When the price $x$ is $4.5$ dollars, the demand $y$ is $500$ units. Find a mathematical model that gives the demand $y$ in terms of the price $x$ in dollars. Your answer is $y=$ [ANS] Approximate the demand when the price is $8.5$ dollars. Your answer is: [ANS]", "answer_v1": [ "4.5*500/x", "264.705882352941" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A company has found that the demand for its product varies inversely as the price of the product. When the price $x$ is $3$ dollars, the demand $y$ is $600$ units. Find a mathematical model that gives the demand $y$ in terms of the price $x$ in dollars. Your answer is $y=$ [ANS] Approximate the demand when the price is $6.5$ dollars. Your answer is: [ANS]", "answer_v2": [ "3*600/x", "276.923076923077" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A company has found that the demand for its product varies inversely as the price of the product. When the price $x$ is $3.5$ dollars, the demand $y$ is $550$ units. Find a mathematical model that gives the demand $y$ in terms of the price $x$ in dollars. Your answer is $y=$ [ANS] Approximate the demand when the price is $7$ dollars. Your answer is: [ANS]", "answer_v3": [ "3.5*550/x", "275" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0429", "subject": "Algebra", "topic": "Variation and power functions", "subtopic": "Inverse variation", "level": "2", "keywords": [ "algebra", "function", "inversely proportional" ], "problem_v1": "$z$ varies inversely as $t$. If $z$=7, then $t$=13. Write $z$ as a function of $t$. $z$=[ANS]", "answer_v1": [ "(91)/t" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "$z$ varies inversely as $t$. If $z$=2, then $t$=15. Write $z$ as a function of $t$. $z$=[ANS]", "answer_v2": [ "(30)/t" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "$z$ varies inversely as $t$. If $z$=4, then $t$=13. Write $z$ as a function of $t$. $z$=[ANS]", "answer_v3": [ "(52)/t" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0430", "subject": "Algebra", "topic": "Variation and power functions", "subtopic": "Inverse variation", "level": "2", "keywords": [ "algebra", "inverse variation" ], "problem_v1": "Find the constant of variation $k$ for the stated condition.\n$y$ varies inversely as the square of $x$, and $y=5$ when $x=4.$ Answer: $k=$ [ANS]", "answer_v1": [ "5*4^2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the constant of variation $k$ for the stated condition.\n$y$ varies inversely as the square of $x$, and $y=2$ when $x=6.$ Answer: $k=$ [ANS]", "answer_v2": [ "2*6^2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the constant of variation $k$ for the stated condition.\n$y$ varies inversely as the square of $x$, and $y=3$ when $x=5.$ Answer: $k=$ [ANS]", "answer_v3": [ "3*5^2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0431", "subject": "Algebra", "topic": "Variation and power functions", "subtopic": "Mixed variation", "level": "2", "keywords": [ "Equation", "Variation" ], "problem_v1": "Suppose $p$ varies inversely as the square root of $q$. If $p=4$ when $q=11$, what is $p$ if $q$ is 5? $p$=[ANS]", "answer_v1": [ "5.93295878967653" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose $p$ varies directly as the cube root of $q$. If $p=-11$ when $q=6$, what is $p$ if $q$ is 15? $p$=[ANS]", "answer_v2": [ "-14.929296891272" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose $p$ varies directly as the square root of $q$. If $p=-7$ when $q=9$, what is $p$ if $q$ is 4? $p$=[ANS]", "answer_v3": [ "-4.66666666666667" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0432", "subject": "Algebra", "topic": "Variation and power functions", "subtopic": "Power functions", "level": "1", "keywords": [ "power function", "exponential", "log", "long-run", "dominate" ], "problem_v1": "Can the following function be written in the form of an exponential function or a power function? If not, be sure you can explain why.\ns(x)=\\frac{8}{7x^{-3}} [ANS] A. It is a power function. B. It is an exponential function. C. It is neither an exponential nor power function.", "answer_v1": [ "A" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C" ] ], "problem_v2": "Can the following function be written in the form of an exponential function or a power function? If not, be sure you can explain why.\ns(x)=\\frac{2}{9 \\cdot (-2)^x} [ANS] A. It is an exponential function. B. It is a power function. C. It is neither an exponential nor power function.", "answer_v2": [ "C" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C" ] ], "problem_v3": "Can the following function be written in the form of an exponential function or a power function? If not, be sure you can explain why.\ns(x)=\\frac{4}{7x^{-2}} [ANS] A. It is an exponential function. B. It is a power function. C. It is neither an exponential nor power function.", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C" ] ] }, { "id": "Algebra_0433", "subject": "Algebra", "topic": "Variation and power functions", "subtopic": "Power functions", "level": "4", "keywords": [ "power function", "exponential", "log", "long-run", "dominate" ], "problem_v1": "The table below gives approximate values for three functions, $f$, $g$, and $h$. One is exponential, one is trigonometric, and one is a power function.\n$\\begin{array}{cccccc}\\hline x &-2 &-1 & 0 & 1 & 2 \\\\ \\hline f(x) & 0.056 & 0.167 & 0.5 & 1.5 & 4.5 \\\\ \\hline g(x) & 16 & 2 & 0 &-2 &-16 \\\\ \\hline h(x) & 9 & 8 & 9 & 10 & 9 \\\\ \\hline \\end{array}$\n(a) Which function is the trigonometric function? [ANS] (b) Which function is the exponential function? [ANS] Find a formula for the exponential function $y=$ [ANS]\n(c) Which function is the power function? [ANS] Find a formula for the power function $y=$ [ANS]", "answer_v1": [ "h(x)", "f(x)", "1/2*3^x", "g(x)", "-2*x^3" ], "answer_type_v1": [ "MCS", "MCS", "EX", "MCS", "EX" ], "options_v1": [ [ "f(x)", "g(x)", "h(x)" ], [ "f(x)", "g(x)", "h(x)" ], [], [ "f(x)", "g(x)", "h(x)" ], [] ], "problem_v2": "The table below gives approximate values for three functions, $f$, $g$, and $h$. One is exponential, one is trigonometric, and one is a power function.\n$\\begin{array}{cccccc}\\hline x &-2 &-1 & 0 & 1 & 2 \\\\ \\hline f(x) &-12 &-3 & 0 &-3 &-12 \\\\ \\hline g(x) & 8 & 7 & 8 & 9 & 8 \\\\ \\hline h(x) & 2.25 & 0.75 & 0.25 & 0.083 & 0.028 \\\\ \\hline \\end{array}$\n(a) Which function is the trigonometric function? [ANS] (b) Which function is the exponential function? [ANS] Find a formula for the exponential function $y=$ [ANS]\n(c) Which function is the power function? [ANS] Find a formula for the power function $y=$ [ANS]", "answer_v2": [ "g(x)", "h(x)", "1/4*3^(-x)", "f(x)", "-3*x^2" ], "answer_type_v2": [ "MCS", "MCS", "EX", "MCS", "EX" ], "options_v2": [ [ "f(x)", "g(x)", "h(x)" ], [ "f(x)", "g(x)", "h(x)" ], [], [ "f(x)", "g(x)", "h(x)" ], [] ], "problem_v3": "The table below gives approximate values for three functions, $f$, $g$, and $h$. One is exponential, one is trigonometric, and one is a power function.\n$\\begin{array}{cccccc}\\hline x &-2 &-1 & 0 & 1 & 2 \\\\ \\hline f(x) & 5 & 4 & 5 & 6 & 5 \\\\ \\hline g(x) & 0.056 & 0.167 & 0.5 & 1.5 & 4.5 \\\\ \\hline h(x) &-10 &-2.5 & 0 &-2.5 &-10 \\\\ \\hline \\end{array}$\n(a) Which function is the trigonometric function? [ANS] (b) Which function is the exponential function? [ANS] Find a formula for the exponential function $y=$ [ANS]\n(c) Which function is the power function? [ANS] Find a formula for the power function $y=$ [ANS]", "answer_v3": [ "f(x)", "g(x)", "1/2*3^x", "h(x)", "-2.5*x^2" ], "answer_type_v3": [ "MCS", "MCS", "EX", "MCS", "EX" ], "options_v3": [ [ "f(x)", "g(x)", "h(x)" ], [ "f(x)", "g(x)", "h(x)" ], [], [ "f(x)", "g(x)", "h(x)" ], [] ] }, { "id": "Algebra_0434", "subject": "Algebra", "topic": "Variation and power functions", "subtopic": "Power functions", "level": "2", "keywords": [ "power function", "inversely proportional", "proportional" ], "problem_v1": "Is the function $ T(s)=\\left(6 s^{-3} \\right) \\left(e s^{-3} \\right)$ a power function? If it is, write it in the form $T(s)=ks^p$ and enter exact values for $k$ and $p$. If it is not a power function, enter NONE in both blanks. Do not leave any blanks empty. $k=$ [ANS]\n$p=$ [ANS]", "answer_v1": [ "6*e", "-6" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Is the function $ T(s)=\\left(2 s^{-2} \\right) \\left(e s^{-4} \\right)$ a power function? If it is, write it in the form $T(s)=ks^p$ and enter exact values for $k$ and $p$. If it is not a power function, enter NONE in both blanks. Do not leave any blanks empty. $k=$ [ANS]\n$p=$ [ANS]", "answer_v2": [ "2*e", "-6" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Is the function $ T(s)=\\left(3 s^{-3} \\right) \\left(e s^{-4} \\right)$ a power function? If it is, write it in the form $T(s)=ks^p$ and enter exact values for $k$ and $p$. If it is not a power function, enter NONE in both blanks. Do not leave any blanks empty. $k=$ [ANS]\n$p=$ [ANS]", "answer_v3": [ "3*e", "-7" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0435", "subject": "Algebra", "topic": "Variation and power functions", "subtopic": "Power functions", "level": "2", "keywords": [ "power functions" ], "problem_v1": "Write an equation for $y$ in terms of $x$ if $y$ is proportional to the fourth power of $x$, and $y=405$ when $x=3$. [ANS]", "answer_v1": [ "y = 5*x^4" ], "answer_type_v1": [ "EQ" ], "options_v1": [ [] ], "problem_v2": "Write an equation for $y$ in terms of $x$ if $y$ is proportional to the fifth power of $x$, and $y=96$ when $x=2$. [ANS]", "answer_v2": [ "y = 3*x^5" ], "answer_type_v2": [ "EQ" ], "options_v2": [ [] ], "problem_v3": "Write an equation for $y$ in terms of $x$ if $y$ is proportional to the fourth power of $x$, and $y=48$ when $x=2$. [ANS]", "answer_v3": [ "y = 3*x^4" ], "answer_type_v3": [ "EQ" ], "options_v3": [ [] ] }, { "id": "Algebra_0436", "subject": "Algebra", "topic": "Variation and power functions", "subtopic": "Power functions", "level": "5", "keywords": [ "power functions" ], "problem_v1": "The surface area of a hedgehog is given by $f(M)=k M^{2/3}$, where $M$ is the body mass, and the constant of proportionality $k$ is a positive number that is determined by the general body shape of hedgehogs. Which of the following is a true statement? Be sure you can explain your answer in algebraic terms. [ANS] A. The surface area for a hedgehog of body mass 60 kilograms is smaller than for a hedgehog of 70 kilograms. B. The surface area for a hedgehog of body mass 60 kilograms cannot be compared to a hedgehog of 70 kilograms because $k$ is not known. C. The surface area for a hedgehog of body mass 60 kilograms is the same as for a hedgehog of 70 kilograms. D. The surface area for a hedgehog of body mass 60 kilograms is larger than for a hedgehog of 70 kilograms. E. None of the above", "answer_v1": [ "A" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ] ], "problem_v2": "The surface area of a human is given by $f(M)=k M^{2/3}$, where $M$ is the body mass, and the constant of proportionality $k$ is a positive number that is determined by the general body shape of humans. Which of the following is a true statement? Be sure you can explain your answer in algebraic terms. [ANS] A. The surface area for a human of body mass 60 kilograms is the same as for a human of 70 kilograms. B. The surface area for a human of body mass 60 kilograms cannot be compared to a human of 70 kilograms because $k$ is not known. C. The surface area for a human of body mass 60 kilograms is larger than for a human of 70 kilograms. D. The surface area for a human of body mass 60 kilograms is smaller than for a human of 70 kilograms. E. None of the above", "answer_v2": [ "D" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ] ], "problem_v3": "The surface area of a giraffe is given by $f(M)=k M^{2/3}$, where $M$ is the body mass, and the constant of proportionality $k$ is a positive number that is determined by the general body shape of giraffes. Which of the following is a true statement? Be sure you can explain your answer in algebraic terms. [ANS] A. The surface area for a giraffe of body mass 60 kilograms is the same as for a giraffe of 70 kilograms. B. The surface area for a giraffe of body mass 60 kilograms is smaller than for a giraffe of 70 kilograms. C. The surface area for a giraffe of body mass 60 kilograms cannot be compared to a giraffe of 70 kilograms because $k$ is not known. D. The surface area for a giraffe of body mass 60 kilograms is larger than for a giraffe of 70 kilograms. E. None of the above", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ] ] }, { "id": "Algebra_0437", "subject": "Algebra", "topic": "Variation and power functions", "subtopic": "Power functions", "level": "3", "keywords": [ "power expressions" ], "problem_v1": "Are the functions given below power functions? Answer T (true) or F (false).\n[ANS] 1. $y=3x^3+2x^2$ [ANS] 2. $y=2/(x^3)$ [ANS] 3. $y=x^3/2$ [ANS] 4. $y=14x^{12}$ [ANS] 5. $y=\\sqrt{4x^4}$ [ANS] 6. $y=12 \\cdot 14^x$", "answer_v1": [ "F", "T", "T", "T", "T", "F" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v1": [ [ "Power function", "Not a power function" ], [ "Power function", "Not a power function" ], [ "Power function", "Not a power function" ], [ "Power function", "Not a power function" ], [ "Power function", "Not a power function" ], [ "Power function", "Not a power function" ] ], "problem_v2": "Are the functions given below power functions? Answer T (true) or F (false).\n[ANS] 1. $y=14x^{12}$ [ANS] 2. $y=2/(x^3)$ [ANS] 3. $y=x^3/2$ [ANS] 4. $y=12 \\cdot 14^x$ [ANS] 5. $y=\\sqrt{4x^4}$ [ANS] 6. $y=3x^3+2x^2$", "answer_v2": [ "T", "T", "T", "F", "T", "F" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v2": [ [ "Power function", "Not a power function" ], [ "Power function", "Not a power function" ], [ "Power function", "Not a power function" ], [ "Power function", "Not a power function" ], [ "Power function", "Not a power function" ], [ "Power function", "Not a power function" ] ], "problem_v3": "Are the functions given below power functions? Answer T (true) or F (false).\n[ANS] 1. $y=3x^3+2x^2$ [ANS] 2. $y=14x^{12}$ [ANS] 3. $y=12 \\cdot 14^x$ [ANS] 4. $y=x^3/2$ [ANS] 5. $y=2/(x^3)$ [ANS] 6. $y=\\sqrt{4x^4}$", "answer_v3": [ "F", "T", "F", "T", "T", "T" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v3": [ [ "Power function", "Not a power function" ], [ "Power function", "Not a power function" ], [ "Power function", "Not a power function" ], [ "Power function", "Not a power function" ], [ "Power function", "Not a power function" ], [ "Power function", "Not a power function" ] ] }, { "id": "Algebra_0438", "subject": "Algebra", "topic": "Variation and power functions", "subtopic": "Applications of power functions", "level": "4", "keywords": [ "power function", "inversely proportional", "proportional" ], "problem_v1": "A 30-second commercial during Super Bowl XLII in 2008 cost advertisers $2.7$ million. For the first Super Bowl in 1967, an advertiser could have purchased approximately $26.19$ minutes of advertising time for the same amount of money.\n(a) Assuming that advertising cost is proportional to its length of time, find the cost of advertising, in dollars/second, during the 2008 Super Bowl. cost=[ANS] dollars/second. (round to nearest cent and do not enter commas) (b) Assuming that advertising cost is proportional to its length of time, find the cost of advertising, in dollars/second, during the 1967 Super Bowl. cost=[ANS] dollars/second. (round to nearest cent and do not enter commas) (c) How many times more expensive was Super Bowl advertising in 2008 than in 1967? [ANS] times more expensive (round to nearest whole number)", "answer_v1": [ "90000", "1718.21", "52" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A 30-second commercial during Super Bowl XXXVII in 2003 cost advertisers $2.2$ million. For the first Super Bowl in 1967, an advertiser could have purchased approximately $26.19$ minutes of advertising time for the same amount of money.\n(a) Assuming that advertising cost is proportional to its length of time, find the cost of advertising, in dollars/second, during the 2003 Super Bowl. cost=[ANS] dollars/second. (round to nearest cent and do not enter commas) (b) Assuming that advertising cost is proportional to its length of time, find the cost of advertising, in dollars/second, during the 1967 Super Bowl. cost=[ANS] dollars/second. (round to nearest cent and do not enter commas) (c) How many times more expensive was Super Bowl advertising in 2003 than in 1967? [ANS] times more expensive (round to nearest whole number)", "answer_v2": [ "73333.3", "1400.03", "52" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A 30-second commercial during Super Bowl XXXIX in 2005 cost advertisers $2.4$ million. For the first Super Bowl in 1967, an advertiser could have purchased approximately $26.19$ minutes of advertising time for the same amount of money.\n(a) Assuming that advertising cost is proportional to its length of time, find the cost of advertising, in dollars/second, during the 2005 Super Bowl. cost=[ANS] dollars/second. (round to nearest cent and do not enter commas) (b) Assuming that advertising cost is proportional to its length of time, find the cost of advertising, in dollars/second, during the 1967 Super Bowl. cost=[ANS] dollars/second. (round to nearest cent and do not enter commas) (c) How many times more expensive was Super Bowl advertising in 2005 than in 1967? [ANS] times more expensive (round to nearest whole number)", "answer_v3": [ "80000", "1527.3", "52" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0439", "subject": "Algebra", "topic": "Variation and power functions", "subtopic": "Applications of power functions", "level": "4", "keywords": [ "power function", "inversely proportional", "proportional" ], "problem_v1": "A group of friends rent a house at the beach for spring break for a total cost of \\$ $1480$.\n(a) Suppose all the members of the group pay an equal share of the total cost. Is the cost to each person directly or inversely proportional to the number of people sharing the house? Be sure you can explain your reasoning. [ANS]\n(b) Write a formula for the cost $C$ per person as a function of the number of students who share the house, $n$. $C(n)=$ [ANS]\n(c) How many people are needed to share the house if each student wants to pay a maximum of \\$ $95$? [ANS] students", "answer_v1": [ "Inversely proportional", "1480/n", "16" ], "answer_type_v1": [ "MCS", "EX", "NV" ], "options_v1": [ [ "Directly proportional", "Inversely proportional" ], [], [] ], "problem_v2": "A group of friends rent a house at the beach for spring break for a total cost of \\$ $1100$.\n(a) Suppose all the members of the group pay an equal share of the total cost. Is the cost to each person directly or inversely proportional to the number of people sharing the house? Be sure you can explain your reasoning. [ANS]\n(b) Write a formula for the cost $C$ per person as a function of the number of students who share the house, $n$. $C(n)=$ [ANS]\n(c) How many people are needed to share the house if each student wants to pay a maximum of \\$ $80$? [ANS] students", "answer_v2": [ "Inversely proportional", "1100/n", "14" ], "answer_type_v2": [ "MCS", "EX", "NV" ], "options_v2": [ [ "Directly proportional", "Inversely proportional" ], [], [] ], "problem_v3": "A group of friends rent a house at the beach for spring break for a total cost of \\$ $1110$.\n(a) Suppose all the members of the group pay an equal share of the total cost. Is the cost to each person directly or inversely proportional to the number of people sharing the house? Be sure you can explain your reasoning. [ANS]\n(b) Write a formula for the cost $C$ per person as a function of the number of students who share the house, $n$. $C(n)=$ [ANS]\n(c) How many people are needed to share the house if each student wants to pay a maximum of \\$ $85$? [ANS] students", "answer_v3": [ "Inversely proportional", "1110/n", "14" ], "answer_type_v3": [ "MCS", "EX", "NV" ], "options_v3": [ [ "Directly proportional", "Inversely proportional" ], [], [] ] }, { "id": "Algebra_0440", "subject": "Algebra", "topic": "Variation and power functions", "subtopic": "Applications of power functions", "level": "4", "keywords": [ "power function", "inversely proportional", "proportional" ], "problem_v1": "One of Kepler's three laws of planetary motion states that the square of the period, $P$, of a body orbiting the sun is proportional to the cube of its average distance, $d$, from the sun. The Earth has a period of 365 days and its distance from the sun is approximately $93,000,000$ miles.\n(a) Find $P$ as a function of $d$. $P(d)=$ [ANS]\n(b) The planet Neptune has an average distance from the sun of $2,800,000,000$ miles. How many earth days are in a Neptune year--in other words, what is the period of the planet Neptune? [ANS] days (Round to nearest day and do not enter commas in your answer.)", "answer_v1": [ "365*[d/(9.3E+7)]^(3/2)", "60298" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "One of Kepler's three laws of planetary motion states that the square of the period, $P$, of a body orbiting the sun is proportional to the cube of its average distance, $d$, from the sun. The Earth has a period of 365 days and its distance from the sun is approximately $93,000,000$ miles.\n(a) Find $P$ as a function of $d$. $P(d)=$ [ANS]\n(b) The planet Mercury has an average distance from the sun of $36,000,000$ miles. How many earth days are in a Mercury year--in other words, what is the period of the planet Mercury? [ANS] days (Round to nearest day and do not enter commas in your answer.)", "answer_v2": [ "365*[d/(9.3E+7)]^(3/2)", "88" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "One of Kepler's three laws of planetary motion states that the square of the period, $P$, of a body orbiting the sun is proportional to the cube of its average distance, $d$, from the sun. The Earth has a period of 365 days and its distance from the sun is approximately $93,000,000$ miles.\n(a) Find $P$ as a function of $d$. $P(d)=$ [ANS]\n(b) The planet Mars has an average distance from the sun of $142,000,000$ miles. How many earth days are in a Mars year--in other words, what is the period of the planet Mars? [ANS] days (Round to nearest day and do not enter commas in your answer.)", "answer_v3": [ "365*[d/(9.3E+7)]^(3/2)", "689" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0441", "subject": "Algebra", "topic": "Variation and power functions", "subtopic": "Applications of power functions", "level": "3", "keywords": [ "power functions" ], "problem_v1": "When an aircraft takes off, it accelerates until it reaches its takeoff speed $V$. In doing so it uses up a distance $R$ of the runway, where $R$ is proportional to the square of the takeoff speed. If $V$ is measured in miles per hour and $R$ is measured in feet, then $0.1641$ is the constant of proportionality. If an aircraft has a takeoff speed of about $210$ miles per hour, how much runway does it need? [ANS] feet", "answer_v1": [ "7236.81" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "When an aircraft takes off, it accelerates until it reaches its takeoff speed $V$. In doing so it uses up a distance $R$ of the runway, where $R$ is proportional to the square of the takeoff speed. If $V$ is measured in miles per hour and $R$ is measured in feet, then $0.1638$ is the constant of proportionality. If an aircraft has a takeoff speed of about $220$ miles per hour, how much runway does it need? [ANS] feet", "answer_v2": [ "7927.92" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "When an aircraft takes off, it accelerates until it reaches its takeoff speed $V$. In doing so it uses up a distance $R$ of the runway, where $R$ is proportional to the square of the takeoff speed. If $V$ is measured in miles per hour and $R$ is measured in feet, then $0.1639$ is the constant of proportionality. If an aircraft has a takeoff speed of about $215$ miles per hour, how much runway does it need? [ANS] feet", "answer_v3": [ "7576.28" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0442", "subject": "Algebra", "topic": "Variation and power functions", "subtopic": "Applications of power functions", "level": "5", "keywords": [ "power expressions" ], "problem_v1": "An astronaut $r$ thousand miles from the center of the earth weighs $2880/r^2$ pounds. The surface of the earth is 4000 miles from the center.\n(a) If the astronaut is $h$ miles above the surface of the earth, express $r$ as a function of $h$. Enter your answer as an equation, such as $r=5h-1$. [ANS]\n(b) Express the astronaut's weight $w$, in pounds, as a function of $h$. Enter your answer as an equation, such as $w=5h-1$. [ANS]", "answer_v1": [ "r = 4000+h", "w = 2880/[(4000+h)^2]" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "An astronaut $r$ thousand miles from the center of the earth weighs $2800/r^2$ pounds. The surface of the earth is 4000 miles from the center.\n(a) If the astronaut is $h$ miles above the surface of the earth, express $r$ as a function of $h$. Enter your answer as an equation, such as $r=5h-1$. [ANS]\n(b) Express the astronaut's weight $w$, in pounds, as a function of $h$. Enter your answer as an equation, such as $w=5h-1$. [ANS]", "answer_v2": [ "r = 4000+h", "w = 2800/[(4000+h)^2]" ], "answer_type_v2": [ "EQ", "EQ" ], "options_v2": [ [], [] ], "problem_v3": "An astronaut $r$ thousand miles from the center of the earth weighs $2830/r^2$ pounds. The surface of the earth is 4000 miles from the center.\n(a) If the astronaut is $h$ miles above the surface of the earth, express $r$ as a function of $h$. Enter your answer as an equation, such as $r=5h-1$. [ANS]\n(b) Express the astronaut's weight $w$, in pounds, as a function of $h$. Enter your answer as an equation, such as $w=5h-1$. [ANS]", "answer_v3": [ "r = 4000+h", "w = 2830/[(4000+h)^2]" ], "answer_type_v3": [ "EQ", "EQ" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0443", "subject": "Algebra", "topic": "Systems of equations and inequalities", "subtopic": "Linear systems", "level": "4", "keywords": [ "solve", "system", "equation", "application" ], "problem_v1": "A test has $25$ problems, which are worth a total of $152$ points. There are two types of problems in the test. Each multiple-choice problem is worth $5$ points, and each short-answer problem is worth $8$ points. Write and solve a system equation to answer the following questions.\nThis test has [ANS] multiple-choice problems and [ANS] short-answer problems.", "answer_v1": [ "16", "9" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A test has $18$ problems, which are worth a total of $92$ points. There are two types of problems in the test. Each multiple-choice problem is worth $2$ points, and each short-answer problem is worth $10$ points. Write and solve a system equation to answer the following questions.\nThis test has [ANS] multiple-choice problems and [ANS] short-answer problems.", "answer_v2": [ "11", "7" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A test has $21$ problems, which are worth a total of $111$ points. There are two types of problems in the test. Each multiple-choice problem is worth $3$ points, and each short-answer problem is worth $9$ points. Write and solve a system equation to answer the following questions.\nThis test has [ANS] multiple-choice problems and [ANS] short-answer problems.", "answer_v3": [ "13", "8" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0444", "subject": "Algebra", "topic": "Systems of equations and inequalities", "subtopic": "Linear systems", "level": "3", "keywords": [ "solve", "system", "equation", "application" ], "problem_v1": "You poured some $8\\%$ alcohol solution and some $12\\%$ alcohol solution into a mixing container. Now you have $640$ grams of $10.5 \\%$ alcohol solution. How many grams of $8\\%$ solution and how many grams of $12 \\%$ solution did you pour into the mixing container? Write and solve a system equation to answer the following questions.\nYou mixed [ANS] grams of $8\\%$ solution with [ANS] grams of $12\\%$ solution.", "answer_v1": [ "240", "400" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "You poured some $10\\%$ alcohol solution and some $8\\%$ alcohol solution into a mixing container. Now you have $600$ grams of $8.8 \\%$ alcohol solution. How many grams of $10\\%$ solution and how many grams of $8 \\%$ solution did you pour into the mixing container? Write and solve a system equation to answer the following questions.\nYou mixed [ANS] grams of $10\\%$ solution with [ANS] grams of $8\\%$ solution.", "answer_v2": [ "240", "360" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "You poured some $12\\%$ alcohol solution and some $6\\%$ alcohol solution into a mixing container. Now you have $800$ grams of $9 \\%$ alcohol solution. How many grams of $12\\%$ solution and how many grams of $6 \\%$ solution did you pour into the mixing container? Write and solve a system equation to answer the following questions.\nYou mixed [ANS] grams of $12\\%$ solution with [ANS] grams of $6\\%$ solution.", "answer_v3": [ "400", "400" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0445", "subject": "Algebra", "topic": "Systems of equations and inequalities", "subtopic": "Linear systems", "level": "4", "keywords": [ "solve", "system", "equation", "application" ], "problem_v1": "Town A and Town B were located close to each other, and recently merged into one city. Town A had a population with $8\\%$ African Americans. Town B had a population with $12\\%$ African Americans. After the merge, the new city has a total of $4800$ residents, with $11\\%$ African Americans. How many residents did Town A and Town B used to have? Write and solve a system equation to answer the following questions.\nTown A used to have [ANS] residents, and Town B used to have [ANS] residents.", "answer_v1": [ "1200", "3600" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Town A and Town B were located close to each other, and recently merged into one city. Town A had a population with $6\\%$ Asians. Town B had a population with $8\\%$ Asians. After the merge, the new city has a total of $4000$ residents, with $7.4\\%$ Asians. How many residents did Town A and Town B used to have? Write and solve a system equation to answer the following questions.\nTown A used to have [ANS] residents, and Town B used to have [ANS] residents.", "answer_v2": [ "1200", "2800" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Town A and Town B were located close to each other, and recently merged into one city. Town A had a population with $12\\%$ native Americans. Town B had a population with $6\\%$ native Americans. After the merge, the new city has a total of $4800$ residents, with $7.75\\%$ native Americans. How many residents did Town A and Town B used to have? Write and solve a system equation to answer the following questions.\nTown A used to have [ANS] residents, and Town B used to have [ANS] residents.", "answer_v3": [ "1400", "3400" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0446", "subject": "Algebra", "topic": "Systems of equations and inequalities", "subtopic": "Linear systems", "level": "4", "keywords": [ "logarithms", "exponentials", "exponential growth", "decay", "application" ], "problem_v1": "Country Day's scholarship fund receives a gift of \\$ 174712.77368759. The money is invested in stocks, bonds, and CDs. CDs pay 3.25 \\% interest, bonds pay 3.2 \\% interest, and stocks pay 10.4 \\% interest. Country day invests \\$ 28610.236337025 more in bonds than in CDs. If the annual income form the investments is \\$ 10302.6670865443, how much was invested in each vehicle?\nCountry Day invested \\$ [ANS] in stocks. Country Day invested \\$ [ANS] in bonds. Country Day invested \\$ [ANS] in CDs.", "answer_v1": [ "65161.4312781021", "69080.7893732563", "40470.5530362312" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Country Day's scholarship fund receives a gift of \\$ 131523.663644468. The money is invested in stocks, bonds, and CDs. CDs pay 3.25 \\% interest, bonds pay 5.8 \\% interest, and stocks pay 8.1 \\% interest. Country day invests \\$ 66625.3012644369 more in bonds than in CDs. If the annual income form the investments is \\$ 7693.99078343078, how much was invested in each vehicle?\nCountry Day invested \\$ [ANS] in stocks. Country Day invested \\$ [ANS] in bonds. Country Day invested \\$ [ANS] in CDs.", "answer_v2": [ "24981.0465006158", "86583.9592041448", "19958.6579397078" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Country Day's scholarship fund receives a gift of \\$ 133084.59651155. The money is invested in stocks, bonds, and CDs. CDs pay 3.25 \\% interest, bonds pay 2.8 \\% interest, and stocks pay 9.3 \\% interest. Country day invests \\$ 46265.0999802945 more in bonds than in CDs. If the annual income form the investments is \\$ 6356.84786082015, how much was invested in each vehicle?\nCountry Day invested \\$ [ANS] in stocks. Country Day invested \\$ [ANS] in bonds. Country Day invested \\$ [ANS] in CDs.", "answer_v3": [ "38806.9369131699", "70271.3797893375", "24006.279809043" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0447", "subject": "Algebra", "topic": "Systems of equations and inequalities", "subtopic": "Linear systems", "level": "3", "keywords": [ "algebra" ], "problem_v1": "An airplane travels 2940 miles in 3.5 hours with the aid of a tail wind. It takes 4 hours for the return trip, flying against the same wind. Find the speed of the airplane in the still wind, and the speed of the wind.\nAirplane speed: [ANS] miles per hour\nWind speed: [ANS] miles per hour", "answer_v1": [ "787.5", "52.5" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "An airplane travels 2700 miles in 4.5 hours with the aid of a tail wind. It takes 5 hours for the return trip, flying against the same wind. Find the speed of the airplane in the still wind, and the speed of the wind.\nAirplane speed: [ANS] miles per hour\nWind speed: [ANS] miles per hour", "answer_v2": [ "570", "30" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "An airplane travels 2310 miles in 3.5 hours with the aid of a tail wind. It takes 4 hours for the return trip, flying against the same wind. Find the speed of the airplane in the still wind, and the speed of the wind.\nAirplane speed: [ANS] miles per hour\nWind speed: [ANS] miles per hour", "answer_v3": [ "618.75", "41.25" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0448", "subject": "Algebra", "topic": "Systems of equations and inequalities", "subtopic": "Linear systems", "level": "3", "keywords": [ "algebra" ], "problem_v1": "A company mixes peanuts, cashews, and almonds to obtain 20 oz package worth 5.4 dollars. If peanuts, cashews, and almonds cost 18, 43, and 53 cents per oz respectively, and the amount of cashews is equal to the amount of almonds, how much of each is in the package? [ANS] oz of peanuts, [ANS] oz of cashews, and [ANS] oz of almonds.", "answer_v1": [ "14", "3", "3" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A company mixes peanuts, cashews, and almonds to obtain 22 oz package worth 5.95 dollars. If peanuts, cashews, and almonds cost 10, 45, and 50 cents per oz respectively, and the amount of cashews is equal to the amount of almonds, how much of each is in the package? [ANS] oz of peanuts, [ANS] oz of cashews, and [ANS] oz of almonds.", "answer_v2": [ "12", "5", "5" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A company mixes peanuts, cashews, and almonds to obtain 19 oz package worth 4.51 dollars. If peanuts, cashews, and almonds cost 13, 43, and 51 cents per oz respectively, and the amount of cashews is equal to the amount of almonds, how much of each is in the package? [ANS] oz of peanuts, [ANS] oz of cashews, and [ANS] oz of almonds.", "answer_v3": [ "13", "3", "3" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0449", "subject": "Algebra", "topic": "Systems of equations and inequalities", "subtopic": "Linear systems", "level": "4", "keywords": [ "algebra" ], "problem_v1": "A person is ordered by doctor to take 20 units of vitamin A, 15 units of vitamin D, and 12 units of vitamin E each day. There are three brands of vitamin supplement available: brand X contains 2, 1, and 2 units, brand Y contains 3, 2, and 2 units and brand Z contains 4, 4, and 1 units of vitamin A, D, and E, respectively. How many pills of each brand will provide the required amounts of vitamin? [ANS] pills of brand X, [ANS] pills of brand Y and [ANS] pills of brand Z.", "answer_v1": [ "3", "2", "2" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A person is ordered by doctor to take 15 units of vitamin A, 11 units of vitamin D, and 9 units of vitamin E each day. There are three brands of vitamin supplement available: brand X contains 2, 1, and 2 units, brand Y contains 3, 2, and 2 units and brand Z contains 4, 4, and 1 units of vitamin A, D, and E, respectively. How many pills of each brand will provide the required amounts of vitamin? [ANS] pills of brand X, [ANS] pills of brand Y and [ANS] pills of brand Z.", "answer_v2": [ "1", "3", "1" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A person is ordered by doctor to take 12 units of vitamin A, 9 units of vitamin D, and 7 units of vitamin E each day. There are three brands of vitamin supplement available: brand X contains 2, 1, and 2 units, brand Y contains 3, 2, and 2 units and brand Z contains 4, 4, and 1 units of vitamin A, D, and E, respectively. How many pills of each brand will provide the required amounts of vitamin? [ANS] pills of brand X, [ANS] pills of brand Y and [ANS] pills of brand Z.", "answer_v3": [ "1", "2", "1" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0450", "subject": "Algebra", "topic": "Systems of equations and inequalities", "subtopic": "Linear systems", "level": "4", "keywords": [ "algebra", "application of linear equations", "systems of linear equations", "equations" ], "problem_v1": "During a one-month promotional campaign, Tiger Films gave either a free DVD rental or a 12-serving box of microwave popcorn to new members. It cost the store \\$1 for each free rental and \\$2 for each box of popcorn. A total of 88 new members were signed up and the store's cost for the incentives was \\$131. How many of each incentive were given away? DVD Rentals: [ANS]\nBoxes of Popcorn: [ANS]", "answer_v1": [ "45", "43" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "During a one-month promotional campaign, Tiger Films gave either a free DVD rental or a 12-serving box of microwave popcorn to new members. It cost the store \\$1 for each free rental and \\$2 for each box of popcorn. A total of 89 new members were signed up and the store's cost for the incentives was \\$147. How many of each incentive were given away? DVD Rentals: [ANS]\nBoxes of Popcorn: [ANS]", "answer_v2": [ "31", "58" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "During a one-month promotional campaign, Tiger Films gave either a free DVD rental or a 12-serving box of microwave popcorn to new members. It cost the store \\$1 for each free rental and \\$2 for each box of popcorn. A total of 80 new members were signed up and the store's cost for the incentives was \\$124. How many of each incentive were given away? DVD Rentals: [ANS]\nBoxes of Popcorn: [ANS]", "answer_v3": [ "36", "44" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0451", "subject": "Algebra", "topic": "Systems of equations and inequalities", "subtopic": "Linear systems", "level": "4", "keywords": [ "algebra", "application of linear equations", "systems of linear equations", "equations" ], "problem_v1": "A business shipped 134 packages one day. Customers are charged $\\$3.50$ for each standard-delivery package and $\\$8.50$ for each express-delivery package. Total shipping charges for the day were $\\$689.00$. How many of each kind of package were shipped? Standard-Delivery Packages: [ANS]\nExpress-Delivery Packages: [ANS]", "answer_v1": [ "90", "44" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A business shipped 94 packages one day. Customers are charged $\\$2.50$ for each standard-delivery package and $\\$7.50$ for each express-delivery package. Total shipping charges for the day were $\\$395.00$. How many of each kind of package were shipped? Standard-Delivery Packages: [ANS]\nExpress-Delivery Packages: [ANS]", "answer_v2": [ "62", "32" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A business shipped 106 packages one day. Customers are charged $\\$2.50$ for each standard-delivery package and $\\$8.50$ for each express-delivery package. Total shipping charges for the day were $\\$469.00$. How many of each kind of package were shipped? Standard-Delivery Packages: [ANS]\nExpress-Delivery Packages: [ANS]", "answer_v3": [ "72", "34" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0452", "subject": "Algebra", "topic": "Systems of equations and inequalities", "subtopic": "Linear systems", "level": "4", "keywords": [ "algebra", "application of linear equations", "systems of linear equations", "equations" ], "problem_v1": "A concert venue sold 2125 tickets one evening. Tickets cost $\\$35.00$ for a covered pavilion seat and $\\$20.00$ for a lawn seat. Total receipts were $\\$55{,}625.00$. How many of each type of ticket were sold? Lawn tickets: [ANS]\nPavilion tickets: [ANS]", "answer_v1": [ "1250", "875" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A concert venue sold 1550 tickets one evening. Tickets cost $\\$25.00$ for a covered pavilion seat and $\\$15.00$ for a lawn seat. Total receipts were $\\$28{,}500.00$. How many of each type of ticket were sold? Lawn tickets: [ANS]\nPavilion tickets: [ANS]", "answer_v2": [ "1025", "525" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A concert venue sold 1675 tickets one evening. Tickets cost $\\$30.00$ for a covered pavilion seat and $\\$15.00$ for a lawn seat. Total receipts were $\\$34{,}875.00$. How many of each type of ticket were sold? Lawn tickets: [ANS]\nPavilion tickets: [ANS]", "answer_v3": [ "1025", "650" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0453", "subject": "Algebra", "topic": "Systems of equations and inequalities", "subtopic": "Linear systems", "level": "4", "keywords": [ "algebra", "application of linear equations", "systems of linear equations", "equations" ], "problem_v1": "Justin's boat travels 144 km downstream in 4 hours and it travels 168 km upstream in 7 hours. Find the speed of the boat in still water and the speed of the stream's current. Boat Speed: [ANS] km/h Current Speed: [ANS] km/h", "answer_v1": [ "30", "6" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Justin's boat travels 48 km downstream in 2 hours and it travels 32 km upstream in 4 hours. Find the speed of the boat in still water and the speed of the stream's current. Boat Speed: [ANS] km/h Current Speed: [ANS] km/h", "answer_v2": [ "16", "8" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Justin's boat travels 81 km downstream in 3 hours and it travels 75 km upstream in 5 hours. Find the speed of the boat in still water and the speed of the stream's current. Boat Speed: [ANS] km/h Current Speed: [ANS] km/h", "answer_v3": [ "21", "6" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0454", "subject": "Algebra", "topic": "Systems of equations and inequalities", "subtopic": "Linear systems", "level": "4", "keywords": [ "algebra", "application of linear equations", "systems of linear equations", "equations" ], "problem_v1": "An apple contains 140 calories and 3 g of fiber. A banana contains 150 calories and 4 g of fiber. Set up a system of equations that could be solved to determine the number of apples ($A$) and bananas ($B$) that should be eaten to obtain 5930 calories and 142 g of fiber. Equation 1: [ANS]\nEquation 2: [ANS]", "answer_v1": [ "140*A+150*B = 5930", "3*A+4*B = 142" ], "answer_type_v1": [ "EQ", "EQ" ], "options_v1": [ [], [] ], "problem_v2": "An apple contains 90 calories and 5 g of fiber. A banana contains 110 calories and 3 g of fiber. Set up a system of equations that could be solved to determine the number of apples ($A$) and bananas ($B$) that should be eaten to obtain 3630 calories and 127 g of fiber. Equation 1: [ANS]\nEquation 2: [ANS]", "answer_v2": [ "90*A+110*B = 3630", "5*A+3*B = 127" ], "answer_type_v2": [ "EQ", "EQ" ], "options_v2": [ [], [] ], "problem_v3": "An apple contains 100 calories and 2 g of fiber. A banana contains 130 calories and 3 g of fiber. Set up a system of equations that could be solved to determine the number of apples ($A$) and bananas ($B$) that should be eaten to obtain 3970 calories and 87 g of fiber. Equation 1: [ANS]\nEquation 2: [ANS]", "answer_v3": [ "100*A+130*B = 3970", "2*A+3*B = 87" ], "answer_type_v3": [ "EQ", "EQ" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0456", "subject": "Algebra", "topic": "Functions with fractional exponents and radical functions", "subtopic": "Equations", "level": "3", "keywords": [ "Solve", "Equation", "Root", "algebra", "solve for variable' 'fraction" ], "problem_v1": "Solve for $t$: \\sqrt{t-151}-\\sqrt{t+117}=125\nThe only possible root is $t=$ [ANS]. It is a(n) [ANS] root. (Fill in the second blank with REAL or EXTRANEOUS)", "answer_v1": [ "3924.399184", "EXTRANEOUS" ], "answer_type_v1": [ "NV", "MCS" ], "options_v1": [ [], [ "REAL", "EXTRANEOUS" ] ], "problem_v2": "Solve for $t$: \\sqrt{t-18}-\\sqrt{t+187}=31\nThe only possible root is $t=$ [ANS]. It is a(n) [ANS] root. (Fill in the second blank with REAL or EXTRANEOUS)", "answer_v2": [ "166.68262226847", "EXTRANEOUS" ], "answer_type_v2": [ "NV", "MCS" ], "options_v2": [ [], [ "REAL", "EXTRANEOUS" ] ], "problem_v3": "Solve for $t$: \\sqrt{t-64}-\\sqrt{t+122}=57\nThe only possible root is $t=$ [ANS]. It is a(n) [ANS] root. (Fill in the second blank with REAL or EXTRANEOUS)", "answer_v3": [ "785.912049861496", "EXTRANEOUS" ], "answer_type_v3": [ "NV", "MCS" ], "options_v3": [ [], [ "REAL", "EXTRANEOUS" ] ] }, { "id": "Algebra_0457", "subject": "Algebra", "topic": "Functions with fractional exponents and radical functions", "subtopic": "Equations", "level": "2", "keywords": [ "algebra", "radicals" ], "problem_v1": "Solve the equation 5\\sqrt{n}=n-6. Solutions (separate by commas): $n=$ [ANS]", "answer_v1": [ "(5+1)^2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Solve the equation 2\\sqrt{n}=n-3. Solutions (separate by commas): $n=$ [ANS]", "answer_v2": [ "(2+1)^2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Solve the equation 3\\sqrt{n}=n-4. Solutions (separate by commas): $n=$ [ANS]", "answer_v3": [ "(3+1)^2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0458", "subject": "Algebra", "topic": "Rational equations and functions", "subtopic": "Simplifying", "level": "2", "keywords": [ "Expressions" ], "problem_v1": "Decide whether the expression \\frac{20 y}{4 y+5} can be put in the form \\frac{ax}{a+x}, where $a$ is a constant and $x$ may involve variables. If it can be put in this form, identify $a$ and $x$. If it cannot be put in this form, enter NONE for both $a$ and $x$ below.\n$a=$ [ANS]\n$x=$ [ANS]", "answer_v1": [ "5", "4*y" ], "answer_type_v1": [ "NV", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Decide whether the expression \\frac{12 y}{6 y+2} can be put in the form \\frac{ax}{a+x}, where $a$ is a constant and $x$ may involve variables. If it can be put in this form, identify $a$ and $x$. If it cannot be put in this form, enter NONE for both $a$ and $x$ below.\n$a=$ [ANS]\n$x=$ [ANS]", "answer_v2": [ "2", "6*y" ], "answer_type_v2": [ "NV", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Decide whether the expression \\frac{15 y}{5 y+3} can be put in the form \\frac{ax}{a+x}, where $a$ is a constant and $x$ may involve variables. If it can be put in this form, identify $a$ and $x$. If it cannot be put in this form, enter NONE for both $a$ and $x$ below.\n$a=$ [ANS]\n$x=$ [ANS]", "answer_v3": [ "3", "5*y" ], "answer_type_v3": [ "NV", "EX" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0459", "subject": "Algebra", "topic": "Rational equations and functions", "subtopic": "Rational equations", "level": "2", "keywords": [ "equations", "rational" ], "problem_v1": "Solve the equation for $U$:\n Z=1-\\frac{Q}{U} $U=$ [ANS]", "answer_v1": [ "Q/(1-Z)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Solve the equation for $S$:\n X=1-\\frac{R}{S} $S=$ [ANS]", "answer_v2": [ "R/(1-X)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Solve the equation for $T$:\n X=1-\\frac{Q}{T} $T=$ [ANS]", "answer_v3": [ "Q/(1-X)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0460", "subject": "Algebra", "topic": "Rational equations and functions", "subtopic": "Rational functions", "level": "2", "keywords": [ "rational", "fraction", "numerator", "denominator", "asymptote" ], "problem_v1": "Is the function below a rational function? If it is, write it in reduced form as a ratio of polynomials $ \\frac{p(x)}{q(x)}$. If it is not, enter NONE in both blanks. Do not leave any blanks empty.\n$\\begin{array}{cccc}\\hline & \\frac{x^2}{x-5}-\\frac{6}{x-4}=& & [ANS] [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "x^3-4*x^2-6*x+30", "x^2-9*x+20" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Is the function below a rational function? If it is, write it in reduced form as a ratio of polynomials $ \\frac{p(x)}{q(x)}$. If it is not, enter NONE in both blanks. Do not leave any blanks empty.\n$\\begin{array}{cccc}\\hline & \\frac{x^2}{x-1}-\\frac{3}{x-6}=& & [ANS] [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "x^3-6*x^2-3*x+3", "x^2-7*x+6" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Is the function below a rational function? If it is, write it in reduced form as a ratio of polynomials $ \\frac{p(x)}{q(x)}$. If it is not, enter NONE in both blanks. Do not leave any blanks empty.\n$\\begin{array}{cccc}\\hline & \\frac{x^2}{x-1}-\\frac{4}{x-4}=& & [ANS] [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "x^3-4*x^2-4*x+4", "x^2-5*x+4" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0461", "subject": "Algebra", "topic": "Rational equations and functions", "subtopic": "Graphs of rational functions", "level": "2", "keywords": [ "power function", "inversely proportional", "proportional" ], "problem_v1": "Find:\n(a) $\\lim\\limits_{t\\to-\\infty}\\, (t^{2}+5)=$ [ANS]\n(Enter the word infinity or-infinity if the limit goes to $\\infty$ or $-\\infty$.) (b) $\\lim\\limits_{t\\to\\infty}\\, 4y^{-2}=$ [ANS]\n(Enter the word infinity or-infinity if the limit goes to $\\infty$ or $-\\infty$.)", "answer_v1": [ "infinity", "0" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find:\n(a) $\\lim\\limits_{t\\to-\\infty}\\, (t^{-4}+8)=$ [ANS]\n(Enter the word infinity or-infinity if the limit goes to $\\infty$ or $-\\infty$.) (b) $\\lim\\limits_{t\\to\\infty}\\, 8\\frac{1}{y}=$ [ANS]\n(Enter the word infinity or-infinity if the limit goes to $\\infty$ or $-\\infty$.)", "answer_v2": [ "8", "0" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find:\n(a) $\\lim\\limits_{t\\to-\\infty}\\, (t^{-2}+5)=$ [ANS]\n(Enter the word infinity or-infinity if the limit goes to $\\infty$ or $-\\infty$.) (b) $\\lim\\limits_{t\\to-\\infty}\\, 4y^{-3}=$ [ANS]\n(Enter the word infinity or-infinity if the limit goes to $\\infty$ or $-\\infty$.)", "answer_v3": [ "5", "0" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0462", "subject": "Algebra", "topic": "Rational equations and functions", "subtopic": "Graphs of rational functions", "level": "2", "keywords": [ "Algebra", "Rational Functions" ], "problem_v1": "Graph the functions\nf(x)=\\frac{-x^3+9x^2-20x-30}{x^2-5x} \\qquad \\textrm{and} \\qquad g(x)=-x+4\nin the same window. You may have to adjust the viewing rectangle to answer the following.\nWhat is (are) the similarities between $f(x)$ and $g(x)$? Check all that apply. [ANS] A. they have the same $y$ intercepts B. they have the same asymptotes C. they have the same end behavior D. they have the same $x$ intercepts", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "Graph the functions\nf(x)=\\frac{-x^3+7x^2-10x-18}{x^2-2x} \\qquad \\textrm{and} \\qquad g(x)=-x+5\nin the same window. You may have to adjust the viewing rectangle to answer the following.\nWhat is (are) the similarities between $f(x)$ and $g(x)$? Check all that apply. [ANS] A. they have the same $y$ intercepts B. they have the same asymptotes C. they have the same $x$ intercepts D. they have the same end behavior", "answer_v2": [ "D" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "Graph the functions\nf(x)=\\frac{-x^3+7x^2-12x-20}{x^2-3x} \\qquad \\textrm{and} \\qquad g(x)=-x+4\nin the same window. You may have to adjust the viewing rectangle to answer the following.\nWhat is (are) the similarities between $f(x)$ and $g(x)$? Check all that apply. [ANS] A. they have the same $y$ intercepts B. they have the same end behavior C. they have the same $x$ intercepts D. they have the same asymptotes", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Algebra_0464", "subject": "Algebra", "topic": "Rational equations and functions", "subtopic": "Asymptotes", "level": "3", "keywords": [ "rational", "domain" ], "problem_v1": "Find vertical asymptote(s) of the function ${m(x)={\\frac{3}{x^{2}-7x+10}}}$. Use commas to separate equations if needed. [ANS]", "answer_v1": [ "(x = 5, x = 2)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Find vertical asymptote(s) of the function ${a(x)={-\\frac{7}{x^{2}-81}}}$. Use commas to separate equations if needed. [ANS]", "answer_v2": [ "(x = -9, x = 9)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Find vertical asymptote(s) of the function ${A(x)={-\\frac{5}{x^{2}+2x-8}}}$. Use commas to separate equations if needed. [ANS]", "answer_v3": [ "(x = -4, x = 2)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Algebra_0465", "subject": "Algebra", "topic": "Rational equations and functions", "subtopic": "Asymptotes", "level": "2", "keywords": [ "rational", "fraction", "numerator", "denominator", "asymptote" ], "problem_v1": "Let $ r(x)=\\frac{p(x)}{q(x)}$, where $p$ and $q$ are polynomials of degrees $m$ and $n$ respectively.\n(a) If $r(x) \\to 0$ as $x \\to \\infty$, then [ANS] A. $m=n$ B. $m > n$ C. $m < n$ D. None of the above\n(b) If $r(x) \\to k$ as $x \\to \\infty$, with $k \\not=0$, then [ANS] A. $m=n$ B. $m < n$ C. $m > n$ D. None of the above", "answer_v1": [ "C", "A" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Let $ r(x)=\\frac{p(x)}{q(x)}$, where $p$ and $q$ are polynomials of degrees $m$ and $n$ respectively.\n(a) If $r(x) \\to 0$ as $x \\to \\infty$, then [ANS] A. $m < n$ B. $m > n$ C. $m=n$ D. None of the above\n(b) If $r(x) \\to k$ as $x \\to \\infty$, with $k \\not=0$, then [ANS] A. $m=n$ B. $m > n$ C. $m < n$ D. None of the above", "answer_v2": [ "A", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Let $ r(x)=\\frac{p(x)}{q(x)}$, where $p$ and $q$ are polynomials of degrees $m$ and $n$ respectively.\n(a) If $r(x) \\to 0$ as $x \\to \\infty$, then [ANS] A. $m=n$ B. $m < n$ C. $m > n$ D. None of the above\n(b) If $r(x) \\to k$ as $x \\to \\infty$, with $k \\not=0$, then [ANS] A. $m < n$ B. $m=n$ C. $m > n$ D. None of the above", "answer_v3": [ "B", "B" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Algebra_0466", "subject": "Algebra", "topic": "Rational equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "calculator", "graphing", "intersection" ], "problem_v1": "A drug\u2019s concentration in a patient\u2019s blood, in micrograms per milliliter, can be modeled by the function\n${C(t)={\\frac{8t}{0.06t^{2}+7}}}$ where $t$ is the time, in minutes, since the drug is injected. 1) The drug\u2019s concentration will be [ANS] milligrams per milliliter $25$ minutes since the injection. Round your answers to two decimal places if needed. 2) When the drug\u2019s concentration decreases to $1.86$ milligrams per milliliter, it\u2019s time to give the patient another injection. The nurse should wait for [ANS] minutes before another injection. Round your answers to an integer if needed. 3) If no re-injection is applied, in the long term, the drug\u2019s concentration will be [ANS] milligrams per milliliter.", "answer_v1": [ "4.49438", "70", "0" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A drug\u2019s concentration in a patient\u2019s blood, in micrograms per milliliter, can be modeled by the function\n${C(t)={\\frac{2t}{0.1t^{2}+3.2}}}$ where $t$ is the time, in minutes, since the drug is injected. 1) The drug\u2019s concentration will be [ANS] milligrams per milliliter $15$ minutes since the injection. Round your answers to two decimal places if needed. 2) When the drug\u2019s concentration decreases to $0.22$ milligrams per milliliter, it\u2019s time to give the patient another injection. The nurse should wait for [ANS] minutes before another injection. Round your answers to an integer if needed. 3) If no re-injection is applied, in the long term, the drug\u2019s concentration will be [ANS] milligrams per milliliter.", "answer_v2": [ "1.16732", "90", "0" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A drug\u2019s concentration in a patient\u2019s blood, in micrograms per milliliter, can be modeled by the function\n${C(t)={\\frac{4t}{0.07t^{2}+4.2}}}$ where $t$ is the time, in minutes, since the drug is injected. 1) The drug\u2019s concentration will be [ANS] milligrams per milliliter $20$ minutes since the injection. Round your answers to two decimal places if needed. 2) When the drug\u2019s concentration decreases to $0.86$ milligrams per milliliter, it\u2019s time to give the patient another injection. The nurse should wait for [ANS] minutes before another injection. Round your answers to an integer if needed. 3) If no re-injection is applied, in the long term, the drug\u2019s concentration will be [ANS] milligrams per milliliter.", "answer_v3": [ "2.48447", "65", "0" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0467", "subject": "Algebra", "topic": "Rational equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "rational", "fraction", "numerator", "denominator", "asymptote" ], "problem_v1": "Let $t$ be the time in weeks. At time $t=0$, organic waste is dumped into a pond. The oxygen level in the pond at time $t$ is given by\n$ f(t)=\\frac{t^{2}-t+1}{t^{2}+1}$. Assume $f(0)=1$ is the normal level of oxygen.\n(a) On a separate piece of paper, graph this function. (b) What will happen to the oxygen level in the lake as time goes on? [ANS] (c) Approximately how many weeks must pass before the oxygen level returns to $75$ \\% of its normal level? [ANS] weeks (Round to at least two decimal places.)", "answer_v1": [ "The oxygen level will eventually return to its normal level in the long-run.", "3.732" ], "answer_type_v1": [ "MCS", "NV" ], "options_v1": [ [ "The oxygen level will continue to decrease in the long-run.", "The oxygen level will continue to increase in the long-run.", "The oxygen level will eventually return to its normal level in the long-run.", "It cannot determine based on the given information." ], [] ], "problem_v2": "Let $t$ be the time in weeks. At time $t=0$, organic waste is dumped into a pond. The oxygen level in the pond at time $t$ is given by\n$ f(t)=\\frac{t^{2}-t+1}{t^{2}+1}$. Assume $f(0)=1$ is the normal level of oxygen.\n(a) On a separate piece of paper, graph this function. (b) What will happen to the oxygen level in the lake as time goes on? [ANS] (c) Approximately how many weeks must pass before the oxygen level returns to $60$ \\% of its normal level? [ANS] weeks (Round to at least two decimal places.)", "answer_v2": [ "The oxygen level will eventually return to its normal level in the long-run.", "2" ], "answer_type_v2": [ "MCS", "NV" ], "options_v2": [ [ "The oxygen level will continue to decrease in the long-run.", "The oxygen level will continue to increase in the long-run.", "The oxygen level will eventually return to its normal level in the long-run.", "It cannot determine based on the given information." ], [] ], "problem_v3": "Let $t$ be the time in weeks. At time $t=0$, organic waste is dumped into a pond. The oxygen level in the pond at time $t$ is given by\n$ f(t)=\\frac{t^{2}-t+1}{t^{2}+1}$. Assume $f(0)=1$ is the normal level of oxygen.\n(a) On a separate piece of paper, graph this function. (b) What will happen to the oxygen level in the lake as time goes on? [ANS] (c) Approximately how many weeks must pass before the oxygen level returns to $65$ \\% of its normal level? [ANS] weeks (Round to at least two decimal places.)", "answer_v3": [ "The oxygen level will eventually return to its normal level in the long-run.", "2.449" ], "answer_type_v3": [ "MCS", "NV" ], "options_v3": [ [ "The oxygen level will continue to decrease in the long-run.", "The oxygen level will continue to increase in the long-run.", "The oxygen level will eventually return to its normal level in the long-run.", "It cannot determine based on the given information." ], [] ] }, { "id": "Algebra_0468", "subject": "Algebra", "topic": "Rational equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "rational", "fraction", "numerator", "denominator", "asymptote" ], "problem_v1": "Bronze is an alloy, or mixture, of copper and tin. The alloy initially contains $4$ kg copper and $16$ kg tin. You add $x$ kg of copper to this $20$ kg of alloy. The concentration of copper in the alloy is a function of $x$:\n$f(x)=\\mbox{Concentration of copper} \\=\\frac{\\mbox{Total amount of copper}}{\\mbox{Total amount of alloy}}$.\n(a) Find a formula for $f(x)=$ [ANS]. The statements below correspond to questions (b)-(f):\nA The concentration of the alloy after removing 1 kg of copper. B The initial concentration of the alloy. C The total amount of copper which can be removed in order for the alloy to become pure tin. D The concentration of the alloy after removing 0.6 kg of copper. E The amount of copper needed to be added/removed in order to have an alloy with a copper concentration of 60\\%. F The concentration of the alloy after adding 0.6 kg of copper. G The total amount of the alloy you have if its concentration of copper is 60\\%. H There is no such feature on the graph, so this has no physical meaning.\n(b) Evaluate $\\ f(0.6)=$ [ANS]. Which statement best explains the significance of this expression? [ANS] (enter a letter of the statement A-H) (c) Evaluate $\\ f(0)=$ [ANS]. Which statement best explains the significance of this expression? [ANS] (enter a letter of the statement A-H) (d) Evaluate $\\ f(-1)=$ [ANS]. Which statement best explains the significance of this expression? [ANS] (enter a letter of the statement A-H) (e) Evaluate $\\ f^{-1}(0.6)=$ [ANS]. Which statement best explains the significance of this expression? [ANS] (enter a letter of the statement A-H) (f) Evaluate $\\ f^{-1}(0)=$ [ANS]. Which statement best explains the significance of this expression? [ANS] (enter a letter of the statement A-H) (g) On a separate piece of paper graph $y=f(x)$, labeling all interesting features.", "answer_v1": [ "(x+4)/(x+20)", "4.6/20.6", "F", "1/5", "B", "3/19", "A", "8/0.4", "E", "-4", "C" ], "answer_type_v1": [ "EX", "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS" ], "options_v1": [ [], [], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [], [ "A", "B", "C", "D", "E", "F", "G", "H" ] ], "problem_v2": "Bronze is an alloy, or mixture, of copper and tin. The alloy initially contains $5$ kg copper and $5$ kg tin. You add $x$ kg of copper to this $10$ kg of alloy. The concentration of copper in the alloy is a function of $x$:\n$f(x)=\\mbox{Concentration of copper} \\=\\frac{\\mbox{Total amount of copper}}{\\mbox{Total amount of alloy}}$.\n(a) Find a formula for $f(x)=$ [ANS]. The statements below correspond to questions (b)-(f):\nA The total amount of the alloy you have if its concentration of copper is 30\\%. B The initial concentration of the alloy. C The concentration of the alloy after adding 0.3 kg of copper. D The amount of copper needed to be added/removed in order to have an alloy with a copper concentration of 30\\%. E The total amount of copper which can be removed in order for the alloy to become pure tin. F The concentration of the alloy after removing 3 kg of copper. G The concentration of the alloy after removing 0.3 kg of copper. H There is no such feature on the graph, so this has no physical meaning.\n(b) Evaluate $\\ f(0.3)=$ [ANS]. Which statement best explains the significance of this expression? [ANS] (enter a letter of the statement A-H) (c) Evaluate $\\ f(0)=$ [ANS]. Which statement best explains the significance of this expression? [ANS] (enter a letter of the statement A-H) (d) Evaluate $\\ f(-3)=$ [ANS]. Which statement best explains the significance of this expression? [ANS] (enter a letter of the statement A-H) (e) Evaluate $\\ f^{-1}(0.3)=$ [ANS]. Which statement best explains the significance of this expression? [ANS] (enter a letter of the statement A-H) (f) Evaluate $\\ f^{-1}(0)=$ [ANS]. Which statement best explains the significance of this expression? [ANS] (enter a letter of the statement A-H) (g) On a separate piece of paper graph $y=f(x)$, labeling all interesting features.", "answer_v2": [ "(x+5)/(x+10)", "5.3/10.3", "C", "1/2", "B", "2/7", "F", "-2/0.7", "D", "-5", "E" ], "answer_type_v2": [ "EX", "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS" ], "options_v2": [ [], [], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [], [ "A", "B", "C", "D", "E", "F", "G", "H" ] ], "problem_v3": "Bronze is an alloy, or mixture, of copper and tin. The alloy initially contains $4$ kg copper and $4$ kg tin. You add $x$ kg of copper to this $8$ kg of alloy. The concentration of copper in the alloy is a function of $x$:\n$f(x)=\\mbox{Concentration of copper} \\=\\frac{\\mbox{Total amount of copper}}{\\mbox{Total amount of alloy}}$.\n(a) Find a formula for $f(x)=$ [ANS]. The statements below correspond to questions (b)-(f):\nA The initial concentration of the alloy. B The amount of copper needed to be added/removed in order to have an alloy with a copper concentration of 40\\%. C The total amount of the alloy you have if its concentration of copper is 40\\%. D The concentration of the alloy after removing 0.4 kg of copper. E The total amount of copper which can be removed in order for the alloy to become pure tin. F The concentration of the alloy after adding 0.4 kg of copper. G The concentration of the alloy after removing 2 kg of copper. H There is no such feature on the graph, so this has no physical meaning.\n(b) Evaluate $\\ f(0.4)=$ [ANS]. Which statement best explains the significance of this expression? [ANS] (enter a letter of the statement A-H) (c) Evaluate $\\ f(0)=$ [ANS]. Which statement best explains the significance of this expression? [ANS] (enter a letter of the statement A-H) (d) Evaluate $\\ f(-2)=$ [ANS]. Which statement best explains the significance of this expression? [ANS] (enter a letter of the statement A-H) (e) Evaluate $\\ f^{-1}(0.4)=$ [ANS]. Which statement best explains the significance of this expression? [ANS] (enter a letter of the statement A-H) (f) Evaluate $\\ f^{-1}(0)=$ [ANS]. Which statement best explains the significance of this expression? [ANS] (enter a letter of the statement A-H) (g) On a separate piece of paper graph $y=f(x)$, labeling all interesting features.", "answer_v3": [ "(x+4)/(x+8)", "4.4/8.4", "F", "1/2", "A", "2/6", "G", "-0.8/0.6", "B", "-4", "E" ], "answer_type_v3": [ "EX", "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS" ], "options_v3": [ [], [], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [], [ "A", "B", "C", "D", "E", "F", "G", "H" ] ] }, { "id": "Algebra_0469", "subject": "Algebra", "topic": "Rational equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "calculus", "derivative", "polynomials", "powers", "functions", "rational functions" ], "problem_v1": "Poiseuille's Law gives the rate of flow, $R$, of a gas through a cylindrical pipe in terms of the radius of the pipe, $r$, for a fixed drop in pressure between the two ends of the pipe.\n(a) Find a formula for Poiseuille's Law, given that the rate of flow is proportional to the fourth power of the radius. $R=$ [ANS]\n(Use k for any constant of proportionality you may have in your equation.) (b) If $R=470$ cm ${}^3$/s in a pipe of radius 3 cm for a certain gas, find a formula for the rate of flow of that gas through a pipe of radius $r$ cm. $R=$ [ANS] cm ${}^3$/s (c) What is the rate of flow of the same gas through a pipe with a 5 cm radius? $R=$ [ANS] cm ${}^3$/s", "answer_v1": [ "k*r^4", "470/(3^4)*r^4", "470*(5/3)^4" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Poiseuille's Law gives the rate of flow, $R$, of a gas through a cylindrical pipe in terms of the radius of the pipe, $r$, for a fixed drop in pressure between the two ends of the pipe.\n(a) Find a formula for Poiseuille's Law, given that the rate of flow is proportional to the fourth power of the radius. $R=$ [ANS]\n(Use k for any constant of proportionality you may have in your equation.) (b) If $R=360$ cm ${}^3$/s in a pipe of radius 5 cm for a certain gas, find a formula for the rate of flow of that gas through a pipe of radius $r$ cm. $R=$ [ANS] cm ${}^3$/s (c) What is the rate of flow of the same gas through a pipe with a 6 cm radius? $R=$ [ANS] cm ${}^3$/s", "answer_v2": [ "k*r^4", "360/(5^4)*r^4", "360*(6/5)^4" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Poiseuille's Law gives the rate of flow, $R$, of a gas through a cylindrical pipe in terms of the radius of the pipe, $r$, for a fixed drop in pressure between the two ends of the pipe.\n(a) Find a formula for Poiseuille's Law, given that the rate of flow is proportional to the fourth power of the radius. $R=$ [ANS]\n(Use k for any constant of proportionality you may have in your equation.) (b) If $R=400$ cm ${}^3$/s in a pipe of radius 4 cm for a certain gas, find a formula for the rate of flow of that gas through a pipe of radius $r$ cm. $R=$ [ANS] cm ${}^3$/s (c) What is the rate of flow of the same gas through a pipe with a 5 cm radius? $R=$ [ANS] cm ${}^3$/s", "answer_v3": [ "k*r^4", "400/(4^4)*r^4", "400*(5/4)^4" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0470", "subject": "Algebra", "topic": "Rational equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "algebra" ], "problem_v1": "You can paint a certain room in 9 hours. Your brother can do it in 6 hours. How long does it take the two of you working together? [ANS] hours. Hint: Call $x$ the number of hours it will take the two of you. Think about how much of the room you and your brother each can paint in one hour.", "answer_v1": [ "3.6" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "You can paint a certain room in 3 hours. Your brother can do it in 8 hours. How long does it take the two of you working together? [ANS] hours. Hint: Call $x$ the number of hours it will take the two of you. Think about how much of the room you and your brother each can paint in one hour.", "answer_v2": [ "2.18181818181818" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "You can paint a certain room in 5 hours. Your brother can do it in 6 hours. How long does it take the two of you working together? [ANS] hours. Hint: Call $x$ the number of hours it will take the two of you. Think about how much of the room you and your brother each can paint in one hour.", "answer_v3": [ "2.72727272727273" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0471", "subject": "Algebra", "topic": "Rational equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "calculus", "functions", "models" ], "problem_v1": "Suppose that a rectangle has an area of 75 square meters. Express the perimeter $P$ as a function of the length $x$ of one of the sides. $P(x)$=[ANS]", "answer_v1": [ "2*(x + 75/x)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Suppose that a rectangle has an area of 35 square meters. Express the perimeter $P$ as a function of the length $x$ of one of the sides. $P(x)$=[ANS]", "answer_v2": [ "2*(x + 35/x)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Suppose that a rectangle has an area of 49 square meters. Express the perimeter $P$ as a function of the length $x$ of one of the sides. $P(x)$=[ANS]", "answer_v3": [ "2*(x + 49/x)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0472", "subject": "Algebra", "topic": "Rational equations and functions", "subtopic": "Applications and models", "level": "4", "keywords": [ "Algebra' 'Rational Functions" ], "problem_v1": "The cost in dollars for removing $p$ percent of pollutants from a river in Smith County is C(p)=\\frac{75100 p}{100-p}. Graph this function on your calculator.\nFind the cost of removing 20\\%.\nCost for removing 20\\%, in dollars, is [ANS]\nFind the cost of removing half of the pollutants.\nCost for removing half, in dollars, is [ANS]\nFind the cost of removing all but 5\\% of the pollutants.\nCost for removing all but 5\\%, in dollars, is [ANS]\nWhat is the smallest value $p$ can be?\nThe smallest value $p$ can be is [ANS]\nWhat is the largest integer value $p$ can be?\nThe largest integer value $p$ can be is [ANS]", "answer_v1": [ "18775", "75100", "1426900", "0", "99" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "The cost in dollars for removing $p$ percent of pollutants from a river in Smith County is C(p)=\\frac{61600 p}{100-p}. Graph this function on your calculator.\nFind the cost of removing 20\\%.\nCost for removing 20\\%, in dollars, is [ANS]\nFind the cost of removing half of the pollutants.\nCost for removing half, in dollars, is [ANS]\nFind the cost of removing all but 5\\% of the pollutants.\nCost for removing all but 5\\%, in dollars, is [ANS]\nWhat is the smallest value $p$ can be?\nThe smallest value $p$ can be is [ANS]\nWhat is the largest integer value $p$ can be?\nThe largest integer value $p$ can be is [ANS]", "answer_v2": [ "15400", "61600", "1170400", "0", "99" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "The cost in dollars for removing $p$ percent of pollutants from a river in Smith County is C(p)=\\frac{66300 p}{100-p}. Graph this function on your calculator.\nFind the cost of removing 20\\%.\nCost for removing 20\\%, in dollars, is [ANS]\nFind the cost of removing half of the pollutants.\nCost for removing half, in dollars, is [ANS]\nFind the cost of removing all but 5\\% of the pollutants.\nCost for removing all but 5\\%, in dollars, is [ANS]\nWhat is the smallest value $p$ can be?\nThe smallest value $p$ can be is [ANS]\nWhat is the largest integer value $p$ can be?\nThe largest integer value $p$ can be is [ANS]", "answer_v3": [ "16575", "66300", "1259700", "0", "99" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Algebra_0473", "subject": "Algebra", "topic": "Rational equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "rational", "applications", "equations" ], "problem_v1": "An experienced tinter can tint a car in $5$ hours. A beginning tinter needs $9$ hours to complete the same job. Find how long it takes for the two to do the job together. Answer: [ANS] hours", "answer_v1": [ "3.21429" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "An experienced tinter can tint a car in $2$ hours. A beginning tinter needs $7$ hours to complete the same job. Find how long it takes for the two to do the job together. Answer: [ANS] hours", "answer_v2": [ "1.55556" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "An experienced tinter can tint a car in $3$ hours. A beginning tinter needs $7$ hours to complete the same job. Find how long it takes for the two to do the job together. Answer: [ANS] hours", "answer_v3": [ "2.1" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0474", "subject": "Algebra", "topic": "Rational equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "rational", "applications", "equations" ], "problem_v1": "If Kamina can do a job in $33$ hours and Simon and Kamina working together can do the same job in $13$ hours, find how long it takes Simon to do the job alone. Answer: [ANS] hours", "answer_v1": [ "21.45" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If Kamina can do a job in $15$ hours and Simon and Kamina working together can do the same job in $5$ hours, find how long it takes Simon to do the job alone. Answer: [ANS] hours", "answer_v2": [ "7.5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If Kamina can do a job in $20$ hours and Simon and Kamina working together can do the same job in $8$ hours, find how long it takes Simon to do the job alone. Answer: [ANS] hours", "answer_v3": [ "13.3333" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0475", "subject": "Algebra", "topic": "Rational equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "rational", "applications", "equations" ], "problem_v1": "An amateur cyclist is training for a road race. He rode the first 42-mile portion of his workout at a constant rate. He then reduced his speed by 5 mph for the remaining 27-mile cool-down portion of the workout. Each portion of the workout took equal time. Find the cyclist's rate during the first portion and his rate during the cool-down portion. First Portion: [ANS] mph Cool-Down: [ANS] mph", "answer_v1": [ "14", "9" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "An amateur cyclist is training for a road race. He rode the first 16-mile portion of his workout at a constant rate. He then reduced his speed by 2 mph for the remaining 12-mile cool-down portion of the workout. Each portion of the workout took equal time. Find the cyclist's rate during the first portion and his rate during the cool-down portion. First Portion: [ANS] mph Cool-Down: [ANS] mph", "answer_v2": [ "8", "6" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "An amateur cyclist is training for a road race. He rode the first 20-mile portion of his workout at a constant rate. He then reduced his speed by 4 mph for the remaining 12-mile cool-down portion of the workout. Each portion of the workout took equal time. Find the cyclist's rate during the first portion and his rate during the cool-down portion. First Portion: [ANS] mph Cool-Down: [ANS] mph", "answer_v3": [ "10", "6" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0476", "subject": "Algebra", "topic": "Rational equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "rational", "applications", "equations" ], "problem_v1": "A Level III video editor takes $18$ hours to add special effects to a movie. A Level II video editor takes $26$ hours and a Level I video editor takes $34$ hours. Find how long it would take them to add the special effects if they all work together. Answer: [ANS] hours", "answer_v1": [ "8.10183" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A Level III video editor takes $10$ hours to add special effects to a movie. A Level II video editor takes $20$ hours and a Level I video editor takes $25$ hours. Find how long it would take them to add the special effects if they all work together. Answer: [ANS] hours", "answer_v2": [ "5.26316" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A Level III video editor takes $13$ hours to add special effects to a movie. A Level II video editor takes $21$ hours and a Level I video editor takes $27$ hours. Find how long it would take them to add the special effects if they all work together. Answer: [ANS] hours", "answer_v3": [ "6.18892" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0477", "subject": "Algebra", "topic": "Rational equations and functions", "subtopic": "Applications and models", "level": "5", "keywords": [ "rational", "applications", "equations" ], "problem_v1": "It takes Sadie $2$ day(s) more to build a shed than Mable. If they build it together, it would take them ${\\textstyle\\frac{5}{12}}$ day(s). How long would it take each of them working alone? Sadie: [ANS] day(s) Mable: [ANS] day(s)", "answer_v1": [ "2.5", "0.5" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "It takes Sadie $1$ day(s) less to build a shed than Mable. If they build it together, it would take them ${\\textstyle\\frac{6}{5}}$ day(s). How long would it take each of them working alone? Sadie: [ANS] day(s) Mable: [ANS] day(s)", "answer_v2": [ "2", "3" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "It takes Sadie $1$ day(s) less to build a shed than Mable. If they build it together, it would take them ${\\textstyle\\frac{15}{16}}$ day(s). How long would it take each of them working alone? Sadie: [ANS] day(s) Mable: [ANS] day(s)", "answer_v3": [ "1.5", "2.5" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0478", "subject": "Algebra", "topic": "Rational equations and functions", "subtopic": "Applications and models", "level": "2", "keywords": [ "prealgebra", "common core", "proportions" ], "problem_v1": "Three tea bags are required to make a gallon of iced tea. Determine the number x of tea bags which are required to make eight gallons of tea.\nProportion in the form $\\frac{a}{b}=\\frac{x}{d}$ [ANS]\nThe number tea bags needed is $x=$ [ANS]\nWrite a proportion for the phrase. Then, solve. When necessary, round to the nearest hundredth. Don't use spaces when typing in your proportion.", "answer_v1": [ "3/1=x/8", "24" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Four tea bags are required to make a gallon of iced tea. Determine the number x of tea bags which are required to make four gallons of tea.\nProportion in the form $\\frac{a}{b}=\\frac{x}{d}$ [ANS]\nThe number tea bags needed is $x=$ [ANS]\nWrite a proportion for the phrase. Then, solve. When necessary, round to the nearest hundredth. Don't use spaces when typing in your proportion.", "answer_v2": [ "4/1=x/4", "16" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Three tea bags are required to make a gallon of iced tea. Determine the number x of tea bags which are required to make five gallons of tea.\nProportion in the form $\\frac{a}{b}=\\frac{x}{d}$ [ANS]\nThe number tea bags needed is $x=$ [ANS]\nWrite a proportion for the phrase. Then, solve. When necessary, round to the nearest hundredth. Don't use spaces when typing in your proportion.", "answer_v3": [ "3/1=x/5", "15" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0479", "subject": "Algebra", "topic": "Inverse functions", "subtopic": "1-1 functions", "level": "2", "keywords": [ "Function", "One-to-One", "Injective", "Onto", "Surjective" ], "problem_v1": "Determine if each of the following functions from $\\lbrace a,b,c,d \\rbrace$ to itself is one-to-one and/or onto. Check ALL correct answers.\n(a) $f(a)=d, f(b)=a, f(c)=c, f(d)=b$ [ANS] A. onto. B. one-to-one. C. neither one-to-one nor onto.\n$f(a)=d, f(b)=b, f(c)=c, f(d)=d$ [ANS] A. neither one-to-one nor onto. B. onto. C. one-to-one.\n$f(a)=b, f(b)=b, f(c)=d, f(d)=c$ [ANS] A. neither one-to-one nor onto. B. onto. C. one-to-one.", "answer_v1": [ "AB", "A", "A" ], "answer_type_v1": [ "MCM", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v2": "Determine if each of the following functions from $\\lbrace a,b,c,d \\rbrace$ to itself is one-to-one and/or onto. Check ALL correct answers.\n(a) $f(a)=b, f(b)=b, f(c)=d, f(d)=c$ [ANS] A. neither one-to-one nor onto. B. one-to-one. C. onto.\n$f(a)=d, f(b)=b, f(c)=c, f(d)=d$ [ANS] A. one-to-one. B. neither one-to-one nor onto. C. onto.\n$f(a)=b, f(b)=a, f(c)=b, f(d)=c$ [ANS] A. one-to-one. B. onto. C. neither one-to-one nor onto.", "answer_v2": [ "A", "B", "C" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v3": "Determine if each of the following functions from $\\lbrace a,b,c,d \\rbrace$ to itself is one-to-one and/or onto. Check ALL correct answers.\n(a) $f(a)=b, f(b)=a, f(c)=b, f(d)=c$ [ANS] A. one-to-one. B. neither one-to-one nor onto. C. onto.\n$f(a)=c, f(b)=d, f(c)=a$ [ANS] A. onto. B. neither one-to-one nor onto. C. one-to-one.\n$f(a)=d, f(b)=a, f(c)=c, f(d)=b$ [ANS] A. one-to-one. B. neither one-to-one nor onto. C. onto.", "answer_v3": [ "B", "C", "AC" ], "answer_type_v3": [ "MCS", "MCS", "MCM" ], "options_v3": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ] }, { "id": "Algebra_0480", "subject": "Algebra", "topic": "Inverse functions", "subtopic": "1-1 functions", "level": "2", "keywords": [ "function' 'inverse", "functions", "inverse functions", "one-to-one" ], "problem_v1": "(a) If $f$ is one-to-one and $f(8)=9$, then $f^{-1}(9)=$ [ANS] and $(f(8))^{-1}=$ [ANS]. (b) If $g$ is one-to-one and $g(4)=11$, then $g^{-1}(11)=$ [ANS] and $(g(4))^{-1}=$ [ANS].", "answer_v1": [ "8", "0.111111111111111", "4", "0.0909090909090909" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "(a) If $f$ is one-to-one and $f(-13)=14$, then $f^{-1}(14)=$ [ANS] and $(f(-13))^{-1}=$ [ANS]. (b) If $g$ is one-to-one and $g(-11)=6$, then $g^{-1}(6)=$ [ANS] and $(g(-11))^{-1}=$ [ANS].", "answer_v2": [ "-13", "0.0714285714285714", "-11", "0.166666666666667" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "(a) If $f$ is one-to-one and $f(-6)=10$, then $f^{-1}(10)=$ [ANS] and $(f(-6))^{-1}=$ [ANS]. (b) If $g$ is one-to-one and $g(-7)=9$, then $g^{-1}(9)=$ [ANS] and $(g(-7))^{-1}=$ [ANS].", "answer_v3": [ "-6", "0.1", "-7", "0.111111111111111" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Algebra_0481", "subject": "Algebra", "topic": "Inverse functions", "subtopic": "1-1 functions", "level": "4", "keywords": [ "calculus", "composition of functions", "combining functions", "inverse functions" ], "problem_v1": "Are the following functions invertible? 1. $f(x)$ is the volume of $x$ kg of water at 4 degrees C. Is this invertible? [ANS] 2. $f(t)$ is the total accumulated rainfall in inches $t$ minutes into a sudden rainstorm in July, 2005. Is this invertible? [ANS] 3. $f(w)$ is the cost of mailing a letter weighing $w$ grams. Is this invertible? [ANS]", "answer_v1": [ "yes", "yes", "no" ], "answer_type_v1": [ "TF", "TF", "TF" ], "options_v1": [ [ "yes", "no" ], [ "yes", "no" ], [ "yes", "no" ] ], "problem_v2": "Are the following functions invertible? 1. $f(t)$ is the number of customers in Macy's department store at $t$ minutes past noon on December 18, 2000. Is this invertible? [ANS] 2. $f(w)$ is the cost of mailing a letter weighing $w$ grams. Is this invertible? [ANS] 3. $f(n)$ is the number of students in your calculus class whose birthday is on the $n$ th day of the year. Is this invertible? [ANS]", "answer_v2": [ "no", "no", "no" ], "answer_type_v2": [ "TF", "TF", "TF" ], "options_v2": [ [ "yes", "no" ], [ "yes", "no" ], [ "yes", "no" ] ], "problem_v3": "Are the following functions invertible? 1. $f(d)$ is the number of orange woolen hats sold at a department store on the $d$ th day after September 1, 2003. Is this invertible? [ANS] 2. $f(x)$ is the volume of $x$ kg of water at 4 degrees C. Is this invertible? [ANS] 3. $f(n)$ is the number of students in your calculus class whose birthday is on the $n$ th day of the year. Is this invertible? [ANS]", "answer_v3": [ "no", "yes", "no" ], "answer_type_v3": [ "TF", "TF", "TF" ], "options_v3": [ [ "yes", "no" ], [ "yes", "no" ], [ "yes", "no" ] ] }, { "id": "Algebra_0482", "subject": "Algebra", "topic": "Inverse functions", "subtopic": "Finding the inverse function", "level": "2", "keywords": [ "inverse", "function", "inverse function" ], "problem_v1": "Find the inverse function (if it exists) of $h(x)=\\frac{x}{8x+5}$. If the function is not invertible, enter NONE. $h^{-1}(x)=$ [ANS]\n(Write your inverse function in terms of the independent variable $x$.)", "answer_v1": [ "5*x/(1-8*x)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the inverse function (if it exists) of $h(x)=\\frac{x}{2x+7}$. If the function is not invertible, enter NONE. $h^{-1}(x)=$ [ANS]\n(Write your inverse function in terms of the independent variable $x$.)", "answer_v2": [ "7*x/(1-2*x)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the inverse function (if it exists) of $h(x)=\\frac{x}{4x+5}$. If the function is not invertible, enter NONE. $h^{-1}(x)=$ [ANS]\n(Write your inverse function in terms of the independent variable $x$.)", "answer_v3": [ "5*x/(1-4*x)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0483", "subject": "Algebra", "topic": "Inverse functions", "subtopic": "Finding the inverse function", "level": "2", "keywords": [ "functions", "domain", "range", "input", "output", "interval notation" ], "problem_v1": "The cost (in dollars) of producing $x$ air conditioners is $C=g(x)=630+50x$. Find a formula for the inverse function $g^{-1}(C)$. $g^{-1}(C)=$ [ANS]", "answer_v1": [ "(C-630)/50" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "The cost (in dollars) of producing $x$ air conditioners is $C=g(x)=550+60x$. Find a formula for the inverse function $g^{-1}(C)$. $g^{-1}(C)=$ [ANS]", "answer_v2": [ "(C-550)/60" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "The cost (in dollars) of producing $x$ air conditioners is $C=g(x)=580+55x$. Find a formula for the inverse function $g^{-1}(C)$. $g^{-1}(C)=$ [ANS]", "answer_v3": [ "(C-580)/55" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0484", "subject": "Algebra", "topic": "Inverse functions", "subtopic": "Finding the inverse function", "level": "2", "keywords": [ "decomposition", "inverse", "composition", "combinations", "function" ], "problem_v1": "Let $ r(x)=\\frac{8x-6}{7x+7}$. Find and simplify $r^{-1}(x)=$ [ANS]", "answer_v1": [ "(7*x+6)/(8-7*x)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Let $ r(x)=\\frac{2x-9}{3x+4}$. Find and simplify $r^{-1}(x)=$ [ANS]", "answer_v2": [ "(4*x+9)/(2-3*x)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Let $ r(x)=\\frac{4x-6}{5x+5}$. Find and simplify $r^{-1}(x)=$ [ANS]", "answer_v3": [ "(5*x+6)/(4-5*x)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0485", "subject": "Algebra", "topic": "Inverse functions", "subtopic": "Interpreting inverse functions", "level": "5", "keywords": [ "inverse", "function", "inverse function" ], "problem_v1": "Let $C=f(q)=350+0.2 q$ give the cost in dollars to manufacture $q$ kg of a chemical. a) Which of the following statement(s) correctly explain the meaning of $f^{-1}(C)$? Check all that apply. [ANS] A. The number of kgs. of chemical that can be manufactured for each 1 dollar spent. B. The cost of manufacturing $C$ kgs. the chemical. C. The number of kgs. of the chemical someone can purchase with $C$ dollars. D. The cost of manufacturing one kg. of the chemical. E. The number of kgs. of the chemical that can be manufactured with $C$ dollars. F. None of the above\n(b) Find a formula for $f^{-1}(C)=$ [ANS]", "answer_v1": [ "E", "(C-350)/0.2" ], "answer_type_v1": [ "MCS", "EX" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ], [] ], "problem_v2": "Let $C=f(q)=175+0.3 q$ give the cost in dollars to manufacture $q$ kg of a chemical. a) Which of the following statement(s) correctly explain the meaning of $f^{-1}(C)$? Check all that apply. [ANS] A. The cost of manufacturing one kg. of the chemical. B. The number of kgs. of the chemical someone can purchase with $C$ dollars. C. The number of kgs. of the chemical that can be manufactured with $C$ dollars. D. The number of kgs. of chemical that can be manufactured for each 1 dollar spent. E. The cost of manufacturing $C$ kgs. the chemical. F. None of the above\n(b) Find a formula for $f^{-1}(C)=$ [ANS]", "answer_v2": [ "C", "(C-175)/0.3" ], "answer_type_v2": [ "MCS", "EX" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ], [] ], "problem_v3": "Let $C=f(q)=225+0.2 q$ give the cost in dollars to manufacture $q$ kg of a chemical. a) Which of the following statement(s) correctly explain the meaning of $f^{-1}(C)$? Check all that apply. [ANS] A. The cost of manufacturing $C$ kgs. the chemical. B. The number of kgs. of chemical that can be manufactured for each 1 dollar spent. C. The number of kgs. of the chemical that can be manufactured with $C$ dollars. D. The number of kgs. of the chemical someone can purchase with $C$ dollars. E. The cost of manufacturing one kg. of the chemical. F. None of the above\n(b) Find a formula for $f^{-1}(C)=$ [ANS]", "answer_v3": [ "C", "(C-225)/0.2" ], "answer_type_v3": [ "MCS", "EX" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ], [] ] }, { "id": "Algebra_0486", "subject": "Algebra", "topic": "Inverse functions", "subtopic": "Interpreting inverse functions", "level": "2", "keywords": [ "set theory", "floor", "function", "one-to-one", "inverse" ], "problem_v1": "Let $f(x)=\\lfloor x/2 \\rfloor$. We learned that the floor and the ceiling functions are NOT invertible, but we also learned about the set of preimages of any value in the Range, the set of images. Keeping that in mind, give your answer in interval notation if necessary.\n(a) Find $f^{-1}(\\lbrace 5 \\rbrace)$.\nYour answer is [ANS]\n(b) Find $f^{-1}(\\lbrace-4 \\rbrace)$.\nYour answer is [ANS]\n(c) Find $f^{-1}(\\lbrace x \\mid 5\\le x \\le 8 \\rbrace)$.\nYour answer is [ANS]\n(d) Find $f^{-1}(\\lbrace x \\mid-8\\le x \\le-4 \\rbrace)$.\nYour answer is [ANS]", "answer_v1": [ "[10,12)", "[-8,-6)", "[10,18)", "[-16,-6)" ], "answer_type_v1": [ "INT", "INT", "INT", "INT" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let $f(x)=\\lfloor x/2 \\rfloor$. We learned that the floor and the ceiling functions are NOT invertible, but we also learned about the set of preimages of any value in the Range, the set of images. Keeping that in mind, give your answer in interval notation if necessary.\n(a) Find $f^{-1}(\\lbrace 2 \\rbrace)$.\nYour answer is [ANS]\n(b) Find $f^{-1}(\\lbrace-2 \\rbrace)$.\nYour answer is [ANS]\n(c) Find $f^{-1}(\\lbrace x \\mid 2\\le x \\le 9 \\rbrace)$.\nYour answer is [ANS]\n(d) Find $f^{-1}(\\lbrace x \\mid-7\\le x \\le-2 \\rbrace)$.\nYour answer is [ANS]", "answer_v2": [ "[4,6)", "[-4,-2)", "[4,20)", "[-14,-2)" ], "answer_type_v2": [ "INT", "INT", "INT", "INT" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let $f(x)=\\lfloor x/2 \\rfloor$. We learned that the floor and the ceiling functions are NOT invertible, but we also learned about the set of preimages of any value in the Range, the set of images. Keeping that in mind, give your answer in interval notation if necessary.\n(a) Find $f^{-1}(\\lbrace 3 \\rbrace)$.\nYour answer is [ANS]\n(b) Find $f^{-1}(\\lbrace-3 \\rbrace)$.\nYour answer is [ANS]\n(c) Find $f^{-1}(\\lbrace x \\mid 3\\le x \\le 8 \\rbrace)$.\nYour answer is [ANS]\n(d) Find $f^{-1}(\\lbrace x \\mid-8\\le x \\le-3 \\rbrace)$.\nYour answer is [ANS]", "answer_v3": [ "[6,8)", "[-6,-4)", "[6,18)", "[-16,-4)" ], "answer_type_v3": [ "INT", "INT", "INT", "INT" ], "options_v3": [ [], [], [], [] ] }, { "id": "Algebra_0487", "subject": "Algebra", "topic": "Inverse functions", "subtopic": "Interpreting inverse functions", "level": "3", "keywords": [ "inverse functions", "reciprocal functions", "relations" ], "problem_v1": "Are the following statements true or false?\n[ANS] 1. If $f$ and $h$ are reciprocal relations, then the domain of $h$ is a subset of the domain of $f$. [ANS] 2. $f^{-1} (x)=\\frac{1}{f(x)}$ for any relation $f$. [ANS] 3. If $f$ and $h$ are reciprocal relations, then the domain of $f$ is equal to the domain of $h$. [ANS] 4. If $f$ and $g$ are inverse relations, then the range of $f$ is equal to the domain of $g$. [ANS] 5. Given two relations $f$ and $g$, then $f(g(x))=g(f(x))$.", "answer_v1": [ "TRUE", "FALSE", "FALSE", "True", "False" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF" ], "options_v1": [ [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ] ], "problem_v2": "Are the following statements true or false?\n[ANS] 1. If $f$ and $h$ are reciprocal relations, then the domain of $f$ is equal to the domain of $h$. [ANS] 2. $f^{-1} (x)=\\frac{1}{f(x)}$ for any relation $f$. [ANS] 3. Given two relations $f$ and $g$, then $f(g(x))=g(f(x))$. [ANS] 4. If $f$ and $h$ are reciprocal relations, then the domain of $h$ is a subset of the domain of $f$. [ANS] 5. Given two relations $f$ and $g$, then $f(g(x))=g(f(x))$ only if $f$ and $g$ are inverses of each other and both are nicely defined at $x$.", "answer_v2": [ "FALSE", "FALSE", "FALSE", "True", "True" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF" ], "options_v2": [ [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ] ], "problem_v3": "Are the following statements true or false?\n[ANS] 1. If $f$ and $g$ are inverse relations, then the range of $f$ is equal to the domain of $g$. [ANS] 2. Given two relations $f$ and $g$, then $f(g(x))=g(f(x))$ only if $f$ and $g$ are inverses of each other and both are nicely defined at $x$. [ANS] 3. Given two relations $f$ and $g$, then $f(g(x))=g(f(x))$. [ANS] 4. Given two relations $f$ and $g$, then $f(g(x))=g(f(x))$ only if $f$ and $g$ are inverses of each other. [ANS] 5. If $f$ and $h$ are reciprocal relations, then the domain of $h$ is a subset of the domain of $f$.", "answer_v3": [ "TRUE", "TRUE", "FALSE", "False", "True" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF" ], "options_v3": [ [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ] ] }, { "id": "Algebra_0488", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Exponential functions", "level": "2", "keywords": [ "exponential functions", "growth rate", "growth factor", "continuous exponential growth", "e", "compound interest", "graphs of exponential functions" ], "problem_v1": "Find a formula for the exponential function which satisfies the given conditions: g(10)=70 \\quad\\mbox{and}\\quad g(30)=20. $g(x)=$ [ANS]", "answer_v1": [ "70*(20/70)^[(x-10)/20]" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find a formula for the exponential function which satisfies the given conditions: g(10)=40 \\quad\\mbox{and}\\quad g(30)=25. $g(x)=$ [ANS]", "answer_v2": [ "40*(25/40)^[(x-10)/20]" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find a formula for the exponential function which satisfies the given conditions: g(10)=50 \\quad\\mbox{and}\\quad g(30)=20. $g(x)=$ [ANS]", "answer_v3": [ "50*(20/50)^[(x-10)/20]" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0489", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Exponential functions", "level": "2", "keywords": [ "logarithms", "log", "ln", "half-life", "double-time", "continuous growth rate" ], "problem_v1": "Convert the exponential equation to the form $Q=ae^{kt}$. In the blanks, give the corresponding values for the values of $a$ and $k$ once you have rewritten the formula. Give your answer for $k$ as a decimal, not a percent. $Q=22 (0.462)^t$ $a=$ [ANS]\n$k=$ [ANS]", "answer_v1": [ "22", "ln(0.462)" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Convert the exponential equation to the form $Q=ae^{kt}$. In the blanks, give the corresponding values for the values of $a$ and $k$ once you have rewritten the formula. Give your answer for $k$ as a decimal, not a percent. $Q=4 (0.487)^t$ $a=$ [ANS]\n$k=$ [ANS]", "answer_v2": [ "4", "ln(0.487)" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Convert the exponential equation to the form $Q=ae^{kt}$. In the blanks, give the corresponding values for the values of $a$ and $k$ once you have rewritten the formula. Give your answer for $k$ as a decimal, not a percent. $Q=10 (0.462)^t$ $a=$ [ANS]\n$k=$ [ANS]", "answer_v3": [ "10", "ln(0.462)" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0490", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Exponential functions", "level": "2", "keywords": [ "exponential functions", "growth rate", "growth factor" ], "problem_v1": "Suppose $Q=37.6(0.746)^t$. Give the starting value $a$, the growth factor $b$, and the growth rate $r$ if $Q=a \\cdot b^t=a(1+r)^t$. $a=$ [ANS]\n$b=$ [ANS]\n$r=$ [ANS] \\%", "answer_v1": [ "37.6", "0.746", "-25.4" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Suppose $Q=30.8(0.817)^t$. Give the starting value $a$, the growth factor $b$, and the growth rate $r$ if $Q=a \\cdot b^t=a(1+r)^t$. $a=$ [ANS]\n$b=$ [ANS]\n$r=$ [ANS] \\%", "answer_v2": [ "30.8", "0.817", "-18.3" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Suppose $Q=33.1(0.751)^t$. Give the starting value $a$, the growth factor $b$, and the growth rate $r$ if $Q=a \\cdot b^t=a(1+r)^t$. $a=$ [ANS]\n$b=$ [ANS]\n$r=$ [ANS] \\%", "answer_v3": [ "33.1", "0.751", "-24.9" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0491", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Exponential functions", "level": "2", "keywords": [ "calculus", "function", "exponential function", "exponentials" ], "problem_v1": "If you write the function $P=8 e^{2 t}$ in the form $P=P_{0}a^{t}$, then $P_0=$ [ANS], and $a=$ [ANS]. This function represents exponential [ANS].", "answer_v1": [ "8", "e^2", "growth" ], "answer_type_v1": [ "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "growth", "decay", "neither growth nor decay" ] ], "problem_v2": "If you write the function $P=2 e^{4 t}$ in the form $P=P_{0}a^{t}$, then $P_0=$ [ANS], and $a=$ [ANS]. This function represents exponential [ANS].", "answer_v2": [ "2", "e^4", "growth" ], "answer_type_v2": [ "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "growth", "decay", "neither growth nor decay" ] ], "problem_v3": "If you write the function $P=4 e^{2 t}$ in the form $P=P_{0}a^{t}$, then $P_0=$ [ANS], and $a=$ [ANS]. This function represents exponential [ANS].", "answer_v3": [ "4", "e^2", "growth" ], "answer_type_v3": [ "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "growth", "decay", "neither growth nor decay" ] ] }, { "id": "Algebra_0492", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Exponential functions", "level": "2", "keywords": [ "calculus", "functions", "models" ], "problem_v1": "Suppose that the graph of $f(x)=Ca^x$ passes through the points (2, 15.3125) and (4, 46.89453125). Find a formula for $f(x)$.\n$f(x)$=[ANS]", "answer_v1": [ "5*1.75**x" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Suppose that the graph of $f(x)=Ca^x$ passes through the points (1, 1.5) and (5, 0.09375). Find a formula for $f(x)$.\n$f(x)$=[ANS]", "answer_v2": [ "3*0.5**x" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Suppose that the graph of $f(x)=Ca^x$ passes through the points (1, 3) and (4, 1.265625). Find a formula for $f(x)$.\n$f(x)$=[ANS]", "answer_v3": [ "4*0.75**x" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0494", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Exponential functions", "level": "3", "keywords": [ "word problem" ], "problem_v1": "The following numbers represent values of the dependent variable that corresponds to equally spaced values of the independent variable. Classify these sequences as linear, exponential or neither:\n$\\begin{array}{cc}\\hline sequence & classification \\\\ \\hline 1, 2, 4, 8, 16,... & [ANS] \\\\ \\hline-6,-8,-10,-12,-14,... & [ANS] \\\\ \\hline 2, 4, 6, 8, 10,... & [ANS] \\\\ \\hline-1, 0, 2, 5, 9,... & [ANS] \\\\ \\hline 1, 4, 16, 64, 256,... & [ANS] \\\\ \\hline 1, 16, 81, 256, 625,... & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "EXPONENTIAL", "LINEAR", "LINEAR", "neither", "exponential", "neither" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "linear", "exponential", "neither" ], [ "linear", "exponential", "neither" ], [ "linear", "exponential", "neither" ], [ "linear", "exponential", "neither" ], [ "linear", "exponential", "neither" ], [ "linear", "exponential", "neither" ] ], "problem_v2": "The following numbers represent values of the dependent variable that corresponds to equally spaced values of the independent variable. Classify these sequences as linear, exponential or neither:\n$\\begin{array}{cc}\\hline sequence & classification \\\\ \\hline 1, 4, 9, 16, 25,... & [ANS] \\\\ \\hline-3,-2, 0, 3, 7,... & [ANS] \\\\ \\hline-9,-11,-13,-15,-17,... & [ANS] \\\\ \\hline-1,-3,-9,-27,-81,... & [ANS] \\\\ \\hline 5, 10, 15, 20, 25,... & [ANS] \\\\ \\hline 1, 5, 25, 125, 625,... & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "NEITHER", "NEITHER", "LINEAR", "exponential", "linear", "exponential" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "linear", "exponential", "neither" ], [ "linear", "exponential", "neither" ], [ "linear", "exponential", "neither" ], [ "linear", "exponential", "neither" ], [ "linear", "exponential", "neither" ], [ "linear", "exponential", "neither" ] ], "problem_v3": "The following numbers represent values of the dependent variable that corresponds to equally spaced values of the independent variable. Classify these sequences as linear, exponential or neither:\n$\\begin{array}{cc}\\hline sequence & classification \\\\ \\hline-2,-4,-6,-8,-10,... & [ANS] \\\\ \\hline-2,-5,-8,-11,-14,... & [ANS] \\\\ \\hline 1, 4, 9, 16, 25,... & [ANS] \\\\ \\hline 5, 6, 8, 11, 15,... & [ANS] \\\\ \\hline 2, 10, 50, 250, 1250,... & [ANS] \\\\ \\hline 1,-3, 9,-27, 81,... & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "LINEAR", "LINEAR", "NEITHER", "neither", "exponential", "exponential" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "linear", "exponential", "neither" ], [ "linear", "exponential", "neither" ], [ "linear", "exponential", "neither" ], [ "linear", "exponential", "neither" ], [ "linear", "exponential", "neither" ], [ "linear", "exponential", "neither" ] ] }, { "id": "Algebra_0495", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Properties of logarithms", "level": "3", "keywords": [ "algebra", "logarithm", "logarithm" ], "problem_v1": "In each part, find the value of $x$ in simplest form.\n(a) $x=\\log_3 \\left(\\frac{1}{81}\\right)$ $x$=[ANS]\n(b) $x=\\log_{10} \\sqrt[5]{10}$ $x$=[ANS]\n(c) $x=\\log_{10} 0.01$ $x$=[ANS]", "answer_v1": [ "-4", "0.2", "-2" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "In each part, find the value of $x$ in simplest form.\n(a) $x=\\log_3 \\left(\\frac{1}{9}\\right)$ $x$=[ANS]\n(b) $x=\\log_{10} \\sqrt[6]{10}$ $x$=[ANS]\n(c) $x=\\log_{10} 0.1$ $x$=[ANS]", "answer_v2": [ "-2", "0.166666666666667", "-1" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "In each part, find the value of $x$ in simplest form.\n(a) $x=\\log_3 \\left(\\frac{1}{9}\\right)$ $x$=[ANS]\n(b) $x=\\log_{10} \\sqrt[5]{10}$ $x$=[ANS]\n(c) $x=\\log_{10} 0.1$ $x$=[ANS]", "answer_v3": [ "-2", "0.2", "-1" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0496", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Properties of logarithms", "level": "1", "keywords": [ "logarithms" ], "problem_v1": "Rewrite the exponential equation $e^{-2} \\approx 0.135$ in equivalent logarithmic form. There may be more than one correct answer. [ANS] A. $-1 \\approx \\ln(0.135)$ B. $-2 \\approx \\log(0.135)$ C. $-2 \\approx \\ln(0.135)$ D. $0.135 \\approx \\log(-2)$ E. None of the above", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Rewrite the exponential equation $e^{-4} \\approx 0.018$ in equivalent logarithmic form. There may be more than one correct answer. [ANS] A. $-3 \\approx \\ln(0.018)$ B. $-4 \\approx \\log(0.018)$ C. $0.018 \\approx \\log(-4)$ D. $-4 \\approx \\ln(0.018)$ E. None of the above", "answer_v2": [ "D" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Rewrite the exponential equation $e^{-4} \\approx 0.018$ in equivalent logarithmic form. There may be more than one correct answer. [ANS] A. $-3 \\approx \\ln(0.018)$ B. $-4 \\approx \\ln(0.018)$ C. $0.018 \\approx \\log(-4)$ D. $-4 \\approx \\log(0.018)$ E. None of the above", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ] ] }, { "id": "Algebra_0497", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Properties of logarithms", "level": "2", "keywords": [ "logarithms" ], "problem_v1": "Using laws of logarithms, write the expression below using sums and/or differences of logarithmic expressions which do not contain the logarithms of products, quotients, or powers. $ \\log \\left(6 (x^{2}-y^{2}) \\right)=$ [ANS]", "answer_v1": [ "log10(6)+log10(x+y)+log10(x-y)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Using laws of logarithms, write the expression below using sums and/or differences of logarithmic expressions which do not contain the logarithms of products, quotients, or powers. $ \\log \\left(3 (x^{2}-y^{2}) \\right)=$ [ANS]", "answer_v2": [ "log10(3)+log10(x+y)+log10(x-y)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Using laws of logarithms, write the expression below using sums and/or differences of logarithmic expressions which do not contain the logarithms of products, quotients, or powers. $ \\log \\left(4 (x^{2}-y^{2}) \\right)=$ [ANS]", "answer_v3": [ "log10(4)+log10(x+y)+log10(x-y)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0498", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Properties of logarithms", "level": "3", "keywords": [ "logarithms", "natural log", "common log", "log", "ln", "properties of logs" ], "problem_v1": "Using the properties of logarithms, decide whether each equation is true or not.\n[ANS] 1. $\\ln{(A)} \\ln{(B)}=\\ln{(A)}+\\ln{(B)}$ [ANS] 2. $p \\cdot \\ln{(A)}=\\ln{(A^p)}$ [ANS] 3. $\\log{(\\sqrt{A})}=\\frac{1}{2} \\log{(A)}$ [ANS] 4. $\\log{(AB)}=\\log{(A)}+\\log{(B)}$ [ANS] 5. $\\sqrt{\\ln{(A)}}=\\ln{(A^{(1/2)})}$ [ANS] 6. $ \\frac{\\log{(A)}}{\\log{(B)}}=\\log{(A)}-\\log{(B)}$", "answer_v1": [ "F", "T", "T", "T", "F", "F" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v1": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v2": "Using the properties of logarithms, decide whether each equation is true or not.\n[ANS] 1. $\\log{(AB)}=\\log{(A)}+\\log{(B)}$ [ANS] 2. $p \\cdot \\ln{(A)}=\\ln{(A^p)}$ [ANS] 3. $\\log{(\\sqrt{A})}=\\frac{1}{2} \\log{(A)}$ [ANS] 4. $ \\frac{\\log{(A)}}{\\log{(B)}}=\\log{(A)}-\\log{(B)}$ [ANS] 5. $\\sqrt{\\ln{(A)}}=\\ln{(A^{(1/2)})}$ [ANS] 6. $\\ln{(A)} \\ln{(B)}=\\ln{(A)}+\\ln{(B)}$", "answer_v2": [ "T", "T", "T", "F", "F", "F" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v2": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v3": "Using the properties of logarithms, decide whether each equation is true or not.\n[ANS] 1. $\\ln{(A)} \\ln{(B)}=\\ln{(A)}+\\ln{(B)}$ [ANS] 2. $\\log{(AB)}=\\log{(A)}+\\log{(B)}$ [ANS] 3. $ \\frac{\\log{(A)}}{\\log{(B)}}=\\log{(A)}-\\log{(B)}$ [ANS] 4. $\\log{(\\sqrt{A})}=\\frac{1}{2} \\log{(A)}$ [ANS] 5. $p \\cdot \\ln{(A)}=\\ln{(A^p)}$ [ANS] 6. $\\sqrt{\\ln{(A)}}=\\ln{(A^{(1/2)})}$", "answer_v3": [ "F", "T", "F", "T", "T", "F" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v3": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ] }, { "id": "Algebra_0499", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Properties of logarithms", "level": "2", "keywords": [ "Algebra", "Logarithmic", "evaluation", "logarithm" ], "problem_v1": "Evaluate the expression, correct to six decimal places, by the Change of Base Formula and a calculator. $\\log_2 8=$ [ANS]", "answer_v1": [ "3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the expression, correct to six decimal places, by the Change of Base Formula and a calculator. $\\log_2 3=$ [ANS]", "answer_v2": [ "1.58496250072116" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the expression, correct to six decimal places, by the Change of Base Formula and a calculator. $\\log_2 5=$ [ANS]", "answer_v3": [ "2.32192809488736" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0500", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Properties of logarithms", "level": "3", "keywords": [ "Algebra", "Logarithmic", "logarithms", "calculus" ], "problem_v1": "\\ln (r^ {8} s^ {7} \\sqrt[7]{r^ {8} s^ {4}}) is equal to A \\ln r+B \\ln s where $A=$ [ANS] and where $B=$ [ANS]", "answer_v1": [ "9.14285714285714", "7.57142857142857" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "\\ln (r^ {2} s^ {10} \\sqrt[4]{r^ {5} s^ {10}}) is equal to A \\ln r+B \\ln s where $A=$ [ANS] and where $B=$ [ANS]", "answer_v2": [ "3.25", "12.5" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "\\ln (r^ {4} s^ {7} \\sqrt[5]{r^ {6} s^ {3}}) is equal to A \\ln r+B \\ln s where $A=$ [ANS] and where $B=$ [ANS]", "answer_v3": [ "5.2", "7.6" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0501", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Properties of logarithms", "level": "2", "keywords": [ "logarithm", "laws of logarithms" ], "problem_v1": "Using a change of base formula, one can write\n\\frac{\\log_{11} (42)} {\\log_{11}(5)}=\\log_{m}(p), where $m=$ [ANS]\n$p=$ [ANS]", "answer_v1": [ "5", "42" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Using a change of base formula, one can write\n\\frac{\\log_{8} (30)} {\\log_{8}(7)}=\\log_{m}(p), where $m=$ [ANS]\n$p=$ [ANS]", "answer_v2": [ "7", "30" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Using a change of base formula, one can write\n\\frac{\\log_{6} (37)} {\\log_{6}(3)}=\\log_{m}(p), where $m=$ [ANS]\n$p=$ [ANS]", "answer_v3": [ "3", "37" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0502", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Properties of logarithms", "level": "3", "keywords": [ "algebra", "logarithms", "approximation" ], "problem_v1": "Given that $\\log_2 11 \\approx 3.45943$, evaluate the following expression. \\log_2 \\sqrt{11} Answer: [ANS]\nNote: You must use the exact approximation above in order to receive credit for this problem. You must use the exact approximation above in order to receive credit for this problem.", "answer_v1": [ "0.5*3.45943" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Given that $\\log_2 5 \\approx 2.32193$, evaluate the following expression. \\log_2 \\sqrt{5} Answer: [ANS]\nNote: You must use the exact approximation above in order to receive credit for this problem. You must use the exact approximation above in order to receive credit for this problem.", "answer_v2": [ "0.5*2.32193" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Given that $\\log_2 7 \\approx 2.80735$, evaluate the following expression. \\log_2 \\sqrt{7} Answer: [ANS]\nNote: You must use the exact approximation above in order to receive credit for this problem. You must use the exact approximation above in order to receive credit for this problem.", "answer_v3": [ "0.5*2.80735" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0503", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Properties of logarithms", "level": "2", "keywords": [ "algebra", "logarithms", "common logarithm" ], "problem_v1": "Calculate the following common logarithm. \\log 0.0001 Answer: [ANS]\nNote: You cannot use any operations except division (/) and negation (-). You cannot use any operations except division (/) and negation (-).", "answer_v1": [ "-4" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate the following common logarithm. \\log 0.1 Answer: [ANS]\nNote: You cannot use any operations except division (/) and negation (-). You cannot use any operations except division (/) and negation (-).", "answer_v2": [ "-1" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate the following common logarithm. \\log 0.01 Answer: [ANS]\nNote: You cannot use any operations except division (/) and negation (-). You cannot use any operations except division (/) and negation (-).", "answer_v3": [ "-2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0504", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Properties of logarithms", "level": "3", "keywords": [ "algebra", "evaluation", "logarithm", "Logarithmic" ], "problem_v1": "Rewrite the expression \\ln (a+b)+4 \\ln (a-b)-5 \\ln c as a single logarithm $\\ln A$. Then the function A=[ANS]", "answer_v1": [ "(a+b)*(a-b)**4/c**5" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Rewrite the expression \\ln (a+b)+5 \\ln (a-b)-3 \\ln c as a single logarithm $\\ln A$. Then the function A=[ANS]", "answer_v2": [ "(a+b)*(a-b)**5/c**3" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Rewrite the expression \\ln (a+b)+4 \\ln (a-b)-3 \\ln c as a single logarithm $\\ln A$. Then the function A=[ANS]", "answer_v3": [ "(a+b)*(a-b)**4/c**3" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0505", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Logarithmic functions", "level": "1", "keywords": [ "logarithms", "log", "ln", "asymptote", "graph of logarithms" ], "problem_v1": "Find the hydrogen ion concentration for Lemon Juice, with a pH of 2.0. Hint: $pH=-\\log{\\lbrack H^+\\rbrack}$. [ANS] moles per liter", "answer_v1": [ "10^(-2)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the hydrogen ion concentration for Tomatoes, with a pH of 4.5. Hint: $pH=-\\log{\\lbrack H^+\\rbrack}$. [ANS] moles per liter", "answer_v2": [ "10^(-4.5)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the hydrogen ion concentration for Apples, with a pH of 3.0. Hint: $pH=-\\log{\\lbrack H^+\\rbrack}$. [ANS] moles per liter", "answer_v3": [ "10^(-3)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0507", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Exponential and logarithmic equations", "level": "2", "keywords": [ "exponential functions", "graphs of exponential functions" ], "problem_v1": "Solve $p=30 (0.8)^q$ graphically for $q$ if $p=18$. $q=$ [ANS] (give your answer accurate to at least 1 decimal place)", "answer_v1": [ "2.2892" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Solve $p=16 (0.89)^q$ graphically for $q$ if $p=5$. $q=$ [ANS] (give your answer accurate to at least 1 decimal place)", "answer_v2": [ "9.9812" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Solve $p=21 (0.8)^q$ graphically for $q$ if $p=8$. $q=$ [ANS] (give your answer accurate to at least 1 decimal place)", "answer_v3": [ "4.3249" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0508", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Exponential and logarithmic equations", "level": "2", "keywords": [ "logarithms", "log", "ln", "half-life", "double-time", "continuous growth rate" ], "problem_v1": "Use logarithms to find an EXACT solution to the equation below.\n$7 \\cdot 3^t=245$\n$t=$ [ANS] (do NOT approximate your answer)", "answer_v1": [ "log(35)/log(3)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use logarithms to find an EXACT solution to the equation below.\n$7 \\cdot 2^t=70$\n$t=$ [ANS] (do NOT approximate your answer)", "answer_v2": [ "log(10)/log(2)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use logarithms to find an EXACT solution to the equation below.\n$7 \\cdot 2^t=140$\n$t=$ [ANS] (do NOT approximate your answer)", "answer_v3": [ "log(20)/log(2)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0509", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Exponential and logarithmic equations", "level": "2", "keywords": [ "logarithms", "natural log", "common log", "log", "ln", "properties of logs" ], "problem_v1": "Find the exact solution to the equation below. (Do not give a decimal approximation.) (Do not give a decimal approximation.)\n64 e^{5x+3}=22. $x=$ [ANS]", "answer_v1": [ "[ln(22/64)-3]/5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the exact solution to the equation below. (Do not give a decimal approximation.) (Do not give a decimal approximation.)\n42 e^{2x+2}=30. $x=$ [ANS]", "answer_v2": [ "[ln(30/42)-2]/2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the exact solution to the equation below. (Do not give a decimal approximation.) (Do not give a decimal approximation.)\n50 e^{3x+2}=22. $x=$ [ANS]", "answer_v3": [ "[ln(22/50)-2]/3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0510", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Exponential and logarithmic equations", "level": "3", "keywords": [ "Algebra", "Exponential", "Logarithmic", "Applications", "logarithms" ], "problem_v1": "If $e^ {2x}-2 e^ {x}=+15$, then $x=$ [ANS].", "answer_v1": [ "1.6094379124341" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If $e^ {2x}+5 e^ {x}=+6$, then $x=$ [ANS].", "answer_v2": [ "0" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If $e^ {2x}+1 e^ {x}=+6$, then $x=$ [ANS].", "answer_v3": [ "0.693147180559945" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0511", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Exponential and logarithmic equations", "level": "3", "keywords": [ "calculus", "logarithmic functions", "logarithms", "laws of logarithms" ], "problem_v1": "Solve the equation $16=50 (1.6)^{x}$ for $x$ using logs. $x=$ [ANS]", "answer_v1": [ "[ln(16/50)]/[ln(1.6)]" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Solve the equation $3=75 (1.2)^{x}$ for $x$ using logs. $x=$ [ANS]", "answer_v2": [ "[ln(3/75)]/[ln(1.2)]" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Solve the equation $7=55 (1.3)^{x}$ for $x$ using logs. $x=$ [ANS]", "answer_v3": [ "[ln(7/55)]/[ln(1.3)]" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0512", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Exponential and logarithmic equations", "level": "3", "keywords": [ "calculus", "logarithmic functions", "logarithms", "laws of logarithms" ], "problem_v1": "Solve the expression $T h^{t}=V g^{t}$ for $t$ assuming that all other letters are positive constants. $t=$ [ANS]", "answer_v1": [ "[ln(T/V)]/[ln(g/h)]" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Solve the expression $P a^{t}=S k^{t}$ for $t$ assuming that all other letters are positive constants. $t=$ [ANS]", "answer_v2": [ "[ln(P/S)]/[ln(k/a)]" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Solve the expression $R c^{t}=V g^{t}$ for $t$ assuming that all other letters are positive constants. $t=$ [ANS]", "answer_v3": [ "[ln(R/V)]/[ln(g/c)]" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0513", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Exponential and logarithmic equations", "level": "2", "keywords": [ "Functions", "Inverse", "Exponential", "Logarithmic", "Ln", "Log", "Equations" ], "problem_v1": "Solve for $\\small{x}$ without using a calculating utility. Enter your answer as an expression containing $\\small{e}$. If there is more than one answer, enter them as a comma-separated list.\n$\\small{\\ln\\!\\left(x^{4}\\right)=8}$ $\\small{x=}$ [ANS]", "answer_v1": [ "(-7.38906, 7.38906)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Solve for $\\small{x}$ without using a calculating utility. Enter your answer as an expression containing $\\small{e}$. If there is more than one answer, enter them as a comma-separated list.\n$\\small{\\ln\\!\\left(x^{5}\\right)=1}$ $\\small{x=}$ [ANS]", "answer_v2": [ "e^(1/5)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Solve for $\\small{x}$ without using a calculating utility. Enter your answer as an expression containing $\\small{e}$. If there is more than one answer, enter them as a comma-separated list.\n$\\small{\\ln\\!\\left(x^{4}\\right)=4}$ $\\small{x=}$ [ANS]", "answer_v3": [ "(-2.71828, 2.71828)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Algebra_0514", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Exponential and logarithmic equations", "level": "2", "keywords": [ "exponential functions", "growth rate", "growth factor" ], "problem_v1": "Solve the following equation for $r$:\n$51=108(0.988)^r$\n$r=$ [ANS]", "answer_v1": [ "[log(51)-log(108)]/[log(0.988)]" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Solve the following equation for $r$:\n$55=102(0.98)^r$\n$r=$ [ANS]", "answer_v2": [ "[log(55)-log(102)]/[log(0.98)]" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Solve the following equation for $r$:\n$51=104(0.983)^r$\n$r=$ [ANS]", "answer_v3": [ "[log(51)-log(104)]/[log(0.983)]" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0515", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Exponential and logarithmic equations", "level": "3", "keywords": [ "Algebra", "logarithmic" ], "problem_v1": "Use a graphing calculator to find all solutions to the equation\n3^{-x}=x-2\nGive the solutions as a comma separated list, with each value correct to two decimal places. If there are no solutions, enter None. [ANS]", "answer_v1": [ "2.09959521978367" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use a graphing calculator to find all solutions to the equation\n2^{-x}=x-3\nGive the solutions as a comma separated list, with each value correct to two decimal places. If there are no solutions, enter None. [ANS]", "answer_v2": [ "3.11539147327805" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use a graphing calculator to find all solutions to the equation\n2^{-x}=x-2\nGive the solutions as a comma separated list, with each value correct to two decimal places. If there are no solutions, enter None. [ANS]", "answer_v3": [ "2.21533633307741" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0516", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Graphs", "level": "3", "keywords": [ "functions", "inverse functions", "base", "Algebra", "Exponential", "Logarithmic", "Applications" ], "problem_v1": "For each of the following, find the base $b$ if the graph of $y=b^x$ contains the given point. $(2,9)$ $b=$ [ANS]\n$(1,4)$ $b=$ [ANS]\n$(-2,1)$ $b=$ [ANS]\n$(1,3)$ $b=$ [ANS]\n$(-1,0.5)$ $b=$ [ANS]\n$(1,1)$ $b=$ [ANS]\n$(0.5,1.4142135623731)$ $b=$ [ANS]\n$(-1,0.5)$ $b=$ [ANS]\n$(0.5,0.707106781186548)$ $b=$ [ANS]\n$(-3,0.125)$ $b=$ [ANS]", "answer_v1": [ "3", "4", "1", "3", "2", "1", "2", "2", "0.5", "2" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "For each of the following, find the base $b$ if the graph of $y=b^x$ contains the given point. $(-4,0.0016)$ $b=$ [ANS]\n$(-3,0.125)$ $b=$ [ANS]\n$(4,1)$ $b=$ [ANS]\n$(-3,1)$ $b=$ [ANS]\n$(1,0.5)$ $b=$ [ANS]\n$(1,2)$ $b=$ [ANS]\n$(3,1)$ $b=$ [ANS]\n$(-3,1)$ $b=$ [ANS]\n$(1,1)$ $b=$ [ANS]\n$(-2,0.111111111111111)$ $b=$ [ANS]", "answer_v2": [ "5", "2", "1", "1", "0.5", "2", "1", "1", "1", "3" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "For each of the following, find the base $b$ if the graph of $y=b^x$ contains the given point. $(-2,0.111111111111111)$ $b=$ [ANS]\n$(-2,0.111111111111111)$ $b=$ [ANS]\n$(-3,0.125)$ $b=$ [ANS]\n$(3,125)$ $b=$ [ANS]\n$(3,1)$ $b=$ [ANS]\n$(-2,1)$ $b=$ [ANS]\n$(-4,0.0123456790123457)$ $b=$ [ANS]\n$(4,256)$ $b=$ [ANS]\n$(1,0.5)$ $b=$ [ANS]\n$(-2,0.111111111111111)$ $b=$ [ANS]", "answer_v3": [ "3", "3", "2", "5", "1", "1", "3", "4", "0.5", "3" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Algebra_0517", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Graphs", "level": "3", "keywords": [ "algebra", "logarithm", "Exponential", "Logarithmic", "Applications" ], "problem_v1": "The graph of the function $y=\\log_a x$ goes through $(38,-1)$. Then $a=1/$ [ANS]", "answer_v1": [ "38" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The graph of the function $y=\\log_a x$ goes through $(6,-1)$. Then $a=1/$ [ANS]", "answer_v2": [ "6" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The graph of the function $y=\\log_a x$ goes through $(17,-1)$. Then $a=1/$ [ANS]", "answer_v3": [ "17" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0518", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Graphs", "level": "2", "keywords": [ "Logarithmic functions", "functions", "inverse functions", "intercepts" ], "problem_v1": "Find the $x$-and $y$-intercepts of $f(x)=7 \\log_{7}\\left(8x-4\\right)+21$. Write none if such a point does not exist. $x$-intercept: [ANS]\n$y$-intercept: [ANS]", "answer_v1": [ "(0.500364,0)", "none" ], "answer_type_v1": [ "OL", "OE" ], "options_v1": [ [], [] ], "problem_v2": "Find the $x$-and $y$-intercepts of $f(x)=10 \\ln\\left(4-5x\\right)-20$. Write none if such a point does not exist. $x$-intercept: [ANS]\n$y$-intercept: [ANS]", "answer_v2": [ "(-0.677811,0)", "(0,-6.13706)" ], "answer_type_v2": [ "OL", "OL" ], "options_v2": [ [], [] ], "problem_v3": "Find the $x$-and $y$-intercepts of $f(x)=7 \\log_{3}\\left(-6x-5\\right)+21$. Write none if such a point does not exist. $x$-intercept: [ANS]\n$y$-intercept: [ANS]", "answer_v3": [ "(-0.839506,0)", "none" ], "answer_type_v3": [ "OL", "OE" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0519", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - population growth", "level": "2", "keywords": [ "algebra", "exponential growth", "logarithms", "exponentials", "exponential growth", "decay", "Exponential", "Logarithmic", "Applications" ], "problem_v1": "The rat population in a major metropolitan city is given by the formula $ n(t)=73 e^{0.03 t}$ where $t$ is measured in years since 1993 and $n$ is measured in millions.\n(a) What was the rat population in 1993? [ANS]\n(b) What is the rat population going to be in the year 2008? [ANS]", "answer_v1": [ "7.3E+07", "1.14487E+08" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The rat population in a major metropolitan city is given by the formula $ n(t)=25 e^{0.04 t}$ where $t$ is measured in years since 1990 and $n$ is measured in millions.\n(a) What was the rat population in 1990? [ANS]\n(b) What is the rat population going to be in the year 2004? [ANS]", "answer_v2": [ "2.5E+07", "4.37668E+07" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The rat population in a major metropolitan city is given by the formula $ n(t)=42 e^{0.03 t}$ where $t$ is measured in years since 1991 and $n$ is measured in millions.\n(a) What was the rat population in 1991? [ANS]\n(b) What is the rat population going to be in the year 2006? [ANS]", "answer_v3": [ "4.2E+07", "6.58691E+07" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0520", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - population growth", "level": "5", "keywords": [ "exponential functions", "growth rate", "growth factor", "continuous exponential growth", "e", "compound interest", "graphs of exponential functions" ], "problem_v1": "In January 2005, the population of California was 36.8 million and growing at an annual rate of 1.3\\%. Assume that growth continues at the same rate.\n(a) By how much will the population increase between 2005 and 2030? By [ANS] million (to the nearest 0.001 million) (b) By how much will the population increase between 2030 and 2055? By [ANS] million (to the nearest 0.001 million) (c) Explain how you can tell before doing the calculations which of the two answers in parts\n(a) and (b) is larger. Select ALL statements in A-F which are true if more than one is possible. [ANS] A. The calculation in part (b) is larger since both increases are over 25 year periods, but since the graph of the function bends upward, the increase in the later time period is larger. B. The calculation in part (b) is larger since the exponential function is concave up, and the average rate of change is increasing as time goes on. C. The calculation in part (a) is larger since the graph of the function is concave down, and the average rate of change is decreasing as time goes on. D. The calculation in part (a) is larger since the function is exponential, and exponential graphs grow faster at first, and then flatten. E. The two answers are equal since the change in time from 2005 to 2030 is the same as the change in time from 2030 to 2055. Therefore the change in outputs will be the same. F. None of the above", "answer_v1": [ "36.8*1.013^25-36.8", "36.8*1.013^(2*25)-36.8*1.013^25", "AB" ], "answer_type_v1": [ "NV", "NV", "MCM" ], "options_v1": [ [], [], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "In January 2005, the population of California was 36.8 million and growing at an annual rate of 1.3\\%. Assume that growth continues at the same rate.\n(a) By how much will the population increase between 2005 and 2015? By [ANS] million (to the nearest 0.001 million) (b) By how much will the population increase between 2015 and 2025? By [ANS] million (to the nearest 0.001 million) (c) Explain how you can tell before doing the calculations which of the two answers in parts\n(a) and (b) is larger. Select ALL statements in A-F which are true if more than one is possible. [ANS] A. The calculation in part (a) is larger since the graph of the function is concave down, and the average rate of change is decreasing as time goes on. B. The two answers are equal since the change in time from 2005 to 2015 is the same as the change in time from 2015 to 2025. Therefore the change in outputs will be the same. C. The calculation in part (a) is larger since the function is exponential, and exponential graphs grow faster at first, and then flatten. D. The calculation in part (b) is larger since both increases are over 10 year periods, but since the graph of the function bends upward, the increase in the later time period is larger. E. The calculation in part (b) is larger since the exponential function is concave up, and the average rate of change is increasing as time goes on. F. None of the above", "answer_v2": [ "36.8*1.013^10-36.8", "36.8*1.013^(2*10)-36.8*1.013^10", "DE" ], "answer_type_v2": [ "NV", "NV", "MCM" ], "options_v2": [ [], [], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "In January 2005, the population of California was 36.8 million and growing at an annual rate of 1.3\\%. Assume that growth continues at the same rate.\n(a) By how much will the population increase between 2005 and 2020? By [ANS] million (to the nearest 0.001 million) (b) By how much will the population increase between 2020 and 2035? By [ANS] million (to the nearest 0.001 million) (c) Explain how you can tell before doing the calculations which of the two answers in parts\n(a) and (b) is larger. Select ALL statements in A-F which are true if more than one is possible. [ANS] A. The calculation in part (b) is larger since the exponential function is concave up, and the average rate of change is increasing as time goes on. B. The calculation in part (a) is larger since the graph of the function is concave down, and the average rate of change is decreasing as time goes on. C. The calculation in part (a) is larger since the function is exponential, and exponential graphs grow faster at first, and then flatten. D. The two answers are equal since the change in time from 2005 to 2020 is the same as the change in time from 2020 to 2035. Therefore the change in outputs will be the same. E. The calculation in part (b) is larger since both increases are over 15 year periods, but since the graph of the function bends upward, the increase in the later time period is larger. F. None of the above", "answer_v3": [ "36.8*1.013^15-36.8", "36.8*1.013^(2*15)-36.8*1.013^15", "AE" ], "answer_type_v3": [ "NV", "NV", "MCM" ], "options_v3": [ [], [], [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Algebra_0521", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - population growth", "level": "3", "keywords": [ "exponential functions", "annual growth rate", "linear growth" ], "problem_v1": "In 1940, there were about 10 brown tree snakes per square mile on the island of Guam, and in 1987, there were about 10000 per square mile.\n(a) Find an exponential formula for the number, $N$, of brown tree snakes per square mile on Guam $t$ years after $1940$. $N=$ [ANS]\n(b) On average, what was the annual percent increase in the population during this period? [ANS] \\% (Round to nearest 0.01\\%)", "answer_v1": [ "10*1.1583^t", "15.83" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "In 1940, there were about 10 brown tree snakes per square mile on the island of Guam, and in 2035, there were about 15000 per square mile.\n(a) Find an exponential formula for the number, $N$, of brown tree snakes per square mile on Guam $t$ years after $1940$. $N=$ [ANS]\n(b) On average, what was the annual percent increase in the population during this period? [ANS] \\% (Round to nearest 0.01\\%)", "answer_v2": [ "10*1.08^t", "8" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "In 1940, there were about 10 brown tree snakes per square mile on the island of Guam, and in 2006, there were about 10000 per square mile.\n(a) Find an exponential formula for the number, $N$, of brown tree snakes per square mile on Guam $t$ years after $1940$. $N=$ [ANS]\n(b) On average, what was the annual percent increase in the population during this period? [ANS] \\% (Round to nearest 0.01\\%)", "answer_v3": [ "10*1.1103^t", "11.03" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0522", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - population growth", "level": "4", "keywords": [ "exponential functions", "annual growth rate", "linear growth" ], "problem_v1": "On November 27, 1993, the New York Times reported that wildlife biologists have found a direct link between the increase in the human population in Florida and the decline of the local black bear population. From 1953 to 1993, the human population increased, on average, at a rate of 8\\% per year, while the black bear population decreased at a rate of 6\\% per year. In 1953 the black bear population was 11,000.\n(a) The 1993 human population of Florida was 13 million. What was the human population in 1953? [ANS] people (Round to nearest whole person. Do not include commas in your answer.)\n(b) Find the black bear population for 1993. [ANS] bears (Round to nearest whole bear. Do not include commas in your answer.)\n(c) Had this trend continued, when would the black bear population have numbered less than 120? In the year [ANS]", "answer_v1": [ "598402", "926", "2026" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "On November 27, 1993, the New York Times reported that wildlife biologists have found a direct link between the increase in the human population in Florida and the decline of the local black bear population. From 1953 to 1993, the human population increased, on average, at a rate of 8\\% per year, while the black bear population decreased at a rate of 6\\% per year. In 1953 the black bear population was 11,000.\n(a) The 1993 human population of Florida was 13 million. What was the human population in 1953? [ANS] people (Round to nearest whole person. Do not include commas in your answer.)\n(b) Find the black bear population for 1993. [ANS] bears (Round to nearest whole bear. Do not include commas in your answer.)\n(c) Had this trend continued, when would the black bear population have numbered less than 70? In the year [ANS]", "answer_v2": [ "598402", "926", "2034" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "On November 27, 1993, the New York Times reported that wildlife biologists have found a direct link between the increase in the human population in Florida and the decline of the local black bear population. From 1953 to 1993, the human population increased, on average, at a rate of 8\\% per year, while the black bear population decreased at a rate of 6\\% per year. In 1953 the black bear population was 11,000.\n(a) The 1993 human population of Florida was 13 million. What was the human population in 1953? [ANS] people (Round to nearest whole person. Do not include commas in your answer.)\n(b) Find the black bear population for 1993. [ANS] bears (Round to nearest whole bear. Do not include commas in your answer.)\n(c) Had this trend continued, when would the black bear population have numbered less than 90? In the year [ANS]", "answer_v3": [ "598402", "926", "2030" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0523", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - population growth", "level": "2", "keywords": [ "exponential functions", "growth rate", "growth factor" ], "problem_v1": "What is the growth/decay factor per decade for a city whose population grows by 29\\% per decade? The growth/decay factor is $b=$ [ANS]", "answer_v1": [ "1.29" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "What is the growth/decay factor per decade for a city whose population grows by 13\\% per decade? The growth/decay factor is $b=$ [ANS]", "answer_v2": [ "1.13" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "What is the growth/decay factor per decade for a city whose population grows by 18\\% per decade? The growth/decay factor is $b=$ [ANS]", "answer_v3": [ "1.18" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0524", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - population growth", "level": "5", "keywords": [ "calculus", "functions", "models" ], "problem_v1": "There are currently 25 frogs in a (large) pond. The frog population grows exponentially, tripling every 10 days. How long will it take (in days) for there to be 220 frogs in the pond? Time to 220 frogs: [ANS] days The pond's ecosystem can support 1600 frogs. How long until the situation becomes critical? Time to 1600 frogs: [ANS] days", "answer_v1": [ "19.7954432506913", "37.8557852142874" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "There are currently 11 frogs in a (large) pond. The frog population grows exponentially, tripling every 7 days. How long will it take (in days) for there to be 290 frogs in the pond? Time to 290 frogs: [ANS] days The pond's ecosystem can support 1100 frogs. How long until the situation becomes critical? Time to 1100 frogs: [ANS] days", "answer_v2": [ "20.8480278142914", "29.3426458400514" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "There are currently 16 frogs in a (large) pond. The frog population grows exponentially, tripling every 9 days. How long will it take (in days) for there to be 220 frogs in the pond? Time to 220 frogs: [ANS] days The pond's ecosystem can support 1300 frogs. How long until the situation becomes critical? Time to 1300 frogs: [ANS] days", "answer_v3": [ "21.4719511699724", "36.0252454838916" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0525", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - population growth", "level": "2", "keywords": [ "exponential functions", "growth rate", "growth factor" ], "problem_v1": "A population, $P$, of a certain species of fish in a lake begins with $450$ members and one-half disappear every $6$ years. Give the constants $a$, $b$, and $T$ so that the population is represented by a function of the form $P=ab^{-t/T}$, where $t$ is the time in years since the population was first measured.\n$a=$ [ANS]\n$b=$ [ANS]\n$T=$ [ANS]", "answer_v1": [ "450", "2", "6" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A population, $P$, of a certain species of fish in a lake begins with $200$ members and one-half disappear every $7$ years. Give the constants $a$, $b$, and $T$ so that the population is represented by a function of the form $P=ab^{-t/T}$, where $t$ is the time in years since the population was first measured.\n$a=$ [ANS]\n$b=$ [ANS]\n$T=$ [ANS]", "answer_v2": [ "200", "2", "7" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A population, $P$, of a certain species of fish in a lake begins with $300$ members and one-half disappear every $6$ years. Give the constants $a$, $b$, and $T$ so that the population is represented by a function of the form $P=ab^{-t/T}$, where $t$ is the time in years since the population was first measured.\n$a=$ [ANS]\n$b=$ [ANS]\n$T=$ [ANS]", "answer_v3": [ "300", "2", "6" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0526", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - population growth", "level": "4", "keywords": [ "exponential model", "growth" ], "problem_v1": "A city had a population of 7,774 at the begining of 1968 and has been growing at 7.1\\% per year since then.\n(a) Find the size of the city at the beginning of 1993. Answer: [ANS]\n(b) During what year will the population of the city reach 8,240,527? Answer: [ANS]", "answer_v1": [ "43189.8113507122", "2069" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A city had a population of 1,747 at the begining of 1946 and has been growing at 9.6\\% per year since then.\n(a) Find the size of the city at the beginning of 1975. Answer: [ANS]\n(b) During what year will the population of the city reach 23,722,432? Answer: [ANS]", "answer_v2": [ "24934.4030483527", "2049" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A city had a population of 3,821 at the begining of 1952 and has been growing at 7.2\\% per year since then.\n(a) Find the size of the city at the beginning of 1985. Answer: [ANS]\n(b) During what year will the population of the city reach 5,948,215? Answer: [ANS]", "answer_v3": [ "37897.0107091753", "2057" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0527", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - population growth", "level": "4", "keywords": [ "exponential model", "growth" ], "problem_v1": "Consider a population of coyotes whose intristic growth rate is 9.1\\% and whose carrying capacity in a particular habitat patch is given by 155 individuals. For each population size calculate the corresponding actual growth rate.\n$\\begin{array}{cc}\\hline population size & actual growth rate \\\\ \\hline 85 & [ANS]\\% \\\\ \\hline 95 & [ANS]\\% \\\\ \\hline 105 & [ANS]\\% \\\\ \\hline 135 & [ANS]\\% \\\\ \\hline 155 & [ANS]\\% \\\\ \\hline \\end{array}$", "answer_v1": [ "4.10967741935484", "3.52258064516129", "2.93548387096774", "1.1741935483871", "0" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Consider a population of rabbits whose intristic growth rate is 11.6\\% and whose carrying capacity in a particular habitat patch is given by 95 individuals. For each population size calculate the corresponding actual growth rate.\n$\\begin{array}{cc}\\hline population size & actual growth rate \\\\ \\hline 35 & [ANS]\\% \\\\ \\hline 45 & [ANS]\\% \\\\ \\hline 55 & [ANS]\\% \\\\ \\hline 85 & [ANS]\\% \\\\ \\hline 95 & [ANS]\\% \\\\ \\hline \\end{array}$", "answer_v2": [ "7.32631578947368", "6.10526315789474", "4.88421052631579", "1.22105263157895", "0" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Consider a population of elk whose intristic growth rate is 9.2\\% and whose carrying capacity in a particular habitat patch is given by 110 individuals. For each population size calculate the corresponding actual growth rate.\n$\\begin{array}{cc}\\hline population size & actual growth rate \\\\ \\hline 45 & [ANS]\\% \\\\ \\hline 55 & [ANS]\\% \\\\ \\hline 65 & [ANS]\\% \\\\ \\hline 95 & [ANS]\\% \\\\ \\hline 110 & [ANS]\\% \\\\ \\hline \\end{array}$", "answer_v3": [ "5.43636363636364", "4.6", "3.76363636363636", "1.25454545454545", "0" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Algebra_0528", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - radioactive decay", "level": "3", "keywords": [ "logarithms", "log", "ln", "half-life", "double-time", "continuous growth rate" ], "problem_v1": "Find the half-life of an element which decays by 3.416\\% each day. The half-life is [ANS] days.", "answer_v1": [ "[ln(0.5)]/[ln(0.96584)]" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the half-life of an element which decays by 3.402\\% each day. The half-life is [ANS] days.", "answer_v2": [ "[ln(0.5)]/[ln(0.96598)]" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the half-life of an element which decays by 3.407\\% each day. The half-life is [ANS] days.", "answer_v3": [ "[ln(0.5)]/[ln(0.96593)]" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0529", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - radioactive decay", "level": "4", "keywords": [ "Algebra", "Exponential", "Logarithmic", "Applications", "logarithms", "exponentials", "exponential growth", "decay" ], "problem_v1": "You go to the doctor and he gives you 14 milligrams of radioactive dye. After 24 minutes, 6.5 milligrams of dye remain in your system. To leave the doctor's office, you must pass through a radiation detector without sounding the alarm. If the detector will sound the alarm if more than 2 milligrams of the dye are in your system, how long will your visit to the doctor take, assuming you were given the dye as soon as you arrived? Give your answer to the nearest minute.\nYou will spend [ANS] minutes at the doctor's office.", "answer_v1": [ "60.868725888839" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "You go to the doctor and he gives you 17 milligrams of radioactive dye. After 12 minutes, 4.5 milligrams of dye remain in your system. To leave the doctor's office, you must pass through a radiation detector without sounding the alarm. If the detector will sound the alarm if more than 2 milligrams of the dye are in your system, how long will your visit to the doctor take, assuming you were given the dye as soon as you arrived? Give your answer to the nearest minute.\nYou will spend [ANS] minutes at the doctor's office.", "answer_v2": [ "19.3214200658034" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "You go to the doctor and he gives you 14 milligrams of radioactive dye. After 16 minutes, 5 milligrams of dye remain in your system. To leave the doctor's office, you must pass through a radiation detector without sounding the alarm. If the detector will sound the alarm if more than 2 milligrams of the dye are in your system, how long will your visit to the doctor take, assuming you were given the dye as soon as you arrived? Give your answer to the nearest minute.\nYou will spend [ANS] minutes at the doctor's office.", "answer_v3": [ "30.2389036816377" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0530", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - radioactive decay", "level": "4", "keywords": [ "Algebra", "Exponential", "Logarithmic", "Applications", "exponential equation", "modeling" ], "problem_v1": "A wooden artifact from an ancient tomb contains 50 percent of the carbon-14 that is present in living trees. How long ago was the artifact made? (The half-life of carbon-14 is 5730 years.) Your answer is [ANS] years.", "answer_v1": [ "5730" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A wooden artifact from an ancient tomb contains 20 percent of the carbon-14 that is present in living trees. How long ago was the artifact made? (The half-life of carbon-14 is 5730 years.) Your answer is [ANS] years.", "answer_v2": [ "13304.6479837046" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A wooden artifact from an ancient tomb contains 30 percent of the carbon-14 that is present in living trees. How long ago was the artifact made? (The half-life of carbon-14 is 5730 years.) Your answer is [ANS] years.", "answer_v3": [ "9952.81285457236" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0531", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - radioactive decay", "level": "4", "keywords": [ "Algebra", "Exponential", "Logarithmic", "Applications", "logarithms", "exponentials", "exponential growth", "decay" ], "problem_v1": "The half-life of Palladium-100 is 4 days. After 24 days a sample of Palladium-100 has been reduced to a mass of 5 mg. What was the initial mass (in mg) of the sample? [ANS]\nWhat is the mass 7 weeks after the start? [ANS]", "answer_v1": [ "320", "0.0656950324416964" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The half-life of Palladium-100 is 4 days. After 12 days a sample of Palladium-100 has been reduced to a mass of 7 mg. What was the initial mass (in mg) of the sample? [ANS]\nWhat is the mass 4 weeks after the start? [ANS]", "answer_v2": [ "56", "0.4375" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The half-life of Palladium-100 is 4 days. After 16 days a sample of Palladium-100 has been reduced to a mass of 5 mg. What was the initial mass (in mg) of the sample? [ANS]\nWhat is the mass 5 weeks after the start? [ANS]", "answer_v3": [ "80", "0.185813611719175" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0532", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - radioactive decay", "level": "4", "keywords": [ "calculus", "function", "exponential function", "exponentials" ], "problem_v1": "In the early 1960s, radioactive strontium-90 was released during atmospheric testing of nuclear weapons and got into the bones of people alive at the time. If the half-life of strontium-90 is 31 years, what fraction of the strontium-90 absorbed in 1963 remained in people's bones in 1999? fraction=[ANS]\n(Enter your answer as a decimal or fraction.) (Enter your answer as a decimal or fraction.)", "answer_v1": [ "(1/2)^[(1999-1963)/31]" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "In the early 1960s, radioactive strontium-90 was released during atmospheric testing of nuclear weapons and got into the bones of people alive at the time. If the half-life of strontium-90 is 29 years, what fraction of the strontium-90 absorbed in 1965 remained in people's bones in 1995? fraction=[ANS]\n(Enter your answer as a decimal or fraction.) (Enter your answer as a decimal or fraction.)", "answer_v2": [ "(1/2)^[(1995-1965)/29]" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "In the early 1960s, radioactive strontium-90 was released during atmospheric testing of nuclear weapons and got into the bones of people alive at the time. If the half-life of strontium-90 is 28 years, what fraction of the strontium-90 absorbed in 1963 remained in people's bones in 1994? fraction=[ANS]\n(Enter your answer as a decimal or fraction.) (Enter your answer as a decimal or fraction.)", "answer_v3": [ "(1/2)^[(1994-1963)/28]" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0533", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - radioactive decay", "level": "3", "keywords": [ "exponential functions", "growth rate", "growth factor" ], "problem_v1": "The half life of substance A is $21$ years, and substance B decays at a rate of $30$ \\% each decade.\n(a) Find a formula for a function $f(t)$ that gives the amount of substance A, in milligrams, left after $t$ years, given that the initial quantity was $100$ milligrams. $f(t)=$ [ANS]\n(a) Find a formula for a function $g(t)$ that gives the amount of substance B, in milligrams, left after $t$ years, given that the initial quantity was $100$ milligrams. $g(t)=$ [ANS]\n(c) Of which substance is there less in the long term? [ANS] A. There is less of substance B in the long term. B. There is less of substance A in the long term. C. There is no way to tell.", "answer_v1": [ "100*0.967532^t", "100*0.964961^t", "A" ], "answer_type_v1": [ "EX", "EX", "MCS" ], "options_v1": [ [], [], [ "A", "B", "C" ] ], "problem_v2": "The half life of substance A is $15$ years, and substance B decays at a rate of $30$ \\% each decade.\n(a) Find a formula for a function $f(t)$ that gives the amount of substance A, in milligrams, left after $t$ years, given that the initial quantity was $100$ milligrams. $f(t)=$ [ANS]\n(a) Find a formula for a function $g(t)$ that gives the amount of substance B, in milligrams, left after $t$ years, given that the initial quantity was $100$ milligrams. $g(t)=$ [ANS]\n(c) Of which substance is there less in the long term? [ANS] A. There is less of substance B in the long term. B. There is less of substance A in the long term. C. There is no way to tell.", "answer_v2": [ "100*0.954842^t", "100*0.964961^t", "B" ], "answer_type_v2": [ "EX", "EX", "MCS" ], "options_v2": [ [], [], [ "A", "B", "C" ] ], "problem_v3": "The half life of substance A is $17$ years, and substance B decays at a rate of $30$ \\% each decade.\n(a) Find a formula for a function $f(t)$ that gives the amount of substance A, in milligrams, left after $t$ years, given that the initial quantity was $100$ milligrams. $f(t)=$ [ANS]\n(a) Find a formula for a function $g(t)$ that gives the amount of substance B, in milligrams, left after $t$ years, given that the initial quantity was $100$ milligrams. $g(t)=$ [ANS]\n(c) Of which substance is there less in the long term? [ANS] A. There is less of substance B in the long term. B. There is less of substance A in the long term. C. There is no way to tell.", "answer_v3": [ "100*0.960047^t", "100*0.964961^t", "B" ], "answer_type_v3": [ "EX", "EX", "MCS" ], "options_v3": [ [], [], [ "A", "B", "C" ] ] }, { "id": "Algebra_0534", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "4", "keywords": [ "algebra", "exponential growth" ], "problem_v1": "Assume that the number of bacteria follows an exponential growth model: $ P(t)=P_0 e^{kt}$. The count in the bacteria culture was 700 after 15 minutes and 1600 after 40 minutes.\n(a) What was the initial size of the culture? [ANS]\n(b) Find the population after 75 minutes. [ANS]\n(c) How many minutes after the start of the experiment will the population reach 11000? [ANS]\nNote: webwork usually expects four significant digits so don't round your answers.", "answer_v1": [ "426.27", "5090.39", "98.3023" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Assume that the number of bacteria follows an exponential growth model: $ P(t)=P_0 e^{kt}$. The count in the bacteria culture was 100 after 20 minutes and 1100 after 35 minutes.\n(a) What was the initial size of the culture? [ANS]\n(b) Find the population after 120 minutes. [ANS]\n(c) How many minutes after the start of the experiment will the population reach 11000? [ANS]\nNote: webwork usually expects four significant digits so don't round your answers.", "answer_v2": [ "4.08768", "8.7623E+08", "49.4038" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Assume that the number of bacteria follows an exponential growth model: $ P(t)=P_0 e^{kt}$. The count in the bacteria culture was 300 after 15 minutes and 1300 after 35 minutes.\n(a) What was the initial size of the culture? [ANS]\n(b) Find the population after 70 minutes. [ANS]\n(c) How many minutes after the start of the experiment will the population reach 12000? [ANS]\nNote: webwork usually expects four significant digits so don't round your answers.", "answer_v3": [ "99.886", "16919.3", "65.3142" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0535", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "4", "keywords": [ "logarithms", "log", "ln", "asymptote", "graph of logarithms" ], "problem_v1": "The magnitude of an earthquake is measured relative to the strength of a \"standard\" earthquake, whose seismic waves are of size $W_0$. The magnitude, $M$, of an earthquake with seismic waves of size $W$ is defined to be $ M=\\log_{10} \\left(\\frac{W}{W_0} \\right)$. The value $M$ is called the Richter scale rating of the strength of the earthquake.\n(a) Let $M$ and $m$ represent the magnitude of two earthquakes whose seismic waves are of sizes $W$ and $w$, respectively. Using properties of logarithms, find a simplified formula for the difference $M-m$, in terms of $W$ and $w$. $M-m=$ [ANS] (Enter log10 log10 or logten for the base 10 logarithm.)\n(b) The 1989 earthquake in California had a rating of 7.1 on the Richter scale. How many times larger were the seismic waves in the March 2005 earthquake off the coast of Sumatra, which measured 8.8 on the Richter scale? [ANS]", "answer_v1": [ "log10(W/w)", "10^(8.8-7.1)" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The magnitude of an earthquake is measured relative to the strength of a \"standard\" earthquake, whose seismic waves are of size $W_0$. The magnitude, $M$, of an earthquake with seismic waves of size $W$ is defined to be $ M=\\log_{10} \\left(\\frac{W}{W_0} \\right)$. The value $M$ is called the Richter scale rating of the strength of the earthquake.\n(a) Let $M$ and $m$ represent the magnitude of two earthquakes whose seismic waves are of sizes $W$ and $w$, respectively. Using properties of logarithms, find a simplified formula for the difference $M-m$, in terms of $W$ and $w$. $M-m=$ [ANS] (Enter log10 log10 or logten for the base 10 logarithm.)\n(b) The 1989 earthquake in California had a rating of 6.9 on the Richter scale. How many times larger were the seismic waves in the March 2005 earthquake off the coast of Sumatra, which measured 8.9 on the Richter scale? [ANS]", "answer_v2": [ "log10(W/w)", "10^(8.9-6.9)" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The magnitude of an earthquake is measured relative to the strength of a \"standard\" earthquake, whose seismic waves are of size $W_0$. The magnitude, $M$, of an earthquake with seismic waves of size $W$ is defined to be $ M=\\log_{10} \\left(\\frac{W}{W_0} \\right)$. The value $M$ is called the Richter scale rating of the strength of the earthquake.\n(a) Let $M$ and $m$ represent the magnitude of two earthquakes whose seismic waves are of sizes $W$ and $w$, respectively. Using properties of logarithms, find a simplified formula for the difference $M-m$, in terms of $W$ and $w$. $M-m=$ [ANS] (Enter log10 log10 or logten for the base 10 logarithm.)\n(b) The 1989 earthquake in California had a rating of 6.9 on the Richter scale. How many times larger were the seismic waves in the March 2005 earthquake off the coast of Sumatra, which measured 8.8 on the Richter scale? [ANS]", "answer_v3": [ "log10(W/w)", "10^(8.8-6.9)" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0536", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "4", "keywords": [ "exponential functions", "graphs of exponential functions" ], "problem_v1": "Suppose $y$, the number of cases of a disease, is reduced by 11\\% per year.\n(a) If there are initially 10,000 cases, express $y$ as a function of $t$, the number of years elapsed. $y=$ [ANS] (do not enter any commas in your formula) (b) How many cases will there be 6 years from now? [ANS] cases. (c) How long does it take to reduce the number of cases to 1000? [ANS] years", "answer_v1": [ "10000*0.89^t", "4969.81", "19.7589" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Suppose $y$, the number of cases of a disease, is reduced by 5\\% per year.\n(a) If there are initially 10,000 cases, express $y$ as a function of $t$, the number of years elapsed. $y=$ [ANS] (do not enter any commas in your formula) (b) How many cases will there be 8 years from now? [ANS] cases. (c) How long does it take to reduce the number of cases to 1000? [ANS] years", "answer_v2": [ "10000*0.95^t", "6634.2", "44.8906" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Suppose $y$, the number of cases of a disease, is reduced by 7\\% per year.\n(a) If there are initially 10,000 cases, express $y$ as a function of $t$, the number of years elapsed. $y=$ [ANS] (do not enter any commas in your formula) (b) How many cases will there be 6 years from now? [ANS] cases. (c) How long does it take to reduce the number of cases to 1000? [ANS] years", "answer_v3": [ "10000*0.93^t", "6469.9", "31.7289" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0537", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "3", "keywords": [ "logarithms", "log", "ln", "half-life", "double-time", "continuous growth rate" ], "problem_v1": "A rubber ball is dropped onto a hard surface from a height of 8 feet, and it bounces up and down. At each bounce it rises to 90\\% of the height from which it fell.\n(a) Find a formula for $h(n)$, the height in inches reached by the ball on bounce $n$. $h(n)=$ [ANS]\n(b) How high will the ball bounce on the $12^{\\mbox{th}}$ bounce? $h=$ [ANS] inches (c) How many bounces before the ball rises no higher than an inch? [ANS] bounces", "answer_v1": [ "96*0.9^n", "27.1132", "44" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A rubber ball is dropped onto a hard surface from a height of 10 feet, and it bounces up and down. At each bounce it rises to 80\\% of the height from which it fell.\n(a) Find a formula for $h(n)$, the height in inches reached by the ball on bounce $n$. $h(n)=$ [ANS]\n(b) How high will the ball bounce on the $9^{\\mbox{th}}$ bounce? $h=$ [ANS] inches (c) How many bounces before the ball rises no higher than an inch? [ANS] bounces", "answer_v2": [ "120*0.8^n", "16.1061", "22" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A rubber ball is dropped onto a hard surface from a height of 8 feet, and it bounces up and down. At each bounce it rises to 80\\% of the height from which it fell.\n(a) Find a formula for $h(n)$, the height in inches reached by the ball on bounce $n$. $h(n)=$ [ANS]\n(b) How high will the ball bounce on the $9^{\\mbox{th}}$ bounce? $h=$ [ANS] inches (c) How many bounces before the ball rises no higher than an inch? [ANS] bounces", "answer_v3": [ "96*0.8^n", "12.8849", "21" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0538", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "4", "keywords": [ "logarithms", "log", "ln", "half-life", "double-time", "continuous growth rate" ], "problem_v1": "Oil leaks from a tank. At hour $t=0$ there are 350 gallons of oil in the tank. Each hour after that, 5\\% of the oil leaks out.\n(a) What percent of the original 350 gallons has leaked out after 9 hours? [ANS] \\% (b) If $\\ Q(t)=Q_0 e^{kt} \\ $ is the quantity of oil remaining after $t$ hours, find the value of $k$. $k=$ [ANS]\n(c) What does $k$ tell you about the leaking oil? Select all that apply if more than one statement is true [ANS] A. Because it is less than one, we know the amount of oil in the tank is decreasing. B. It tells by what percent of oil decays each hour. C. Because it is negative, we know the amount of oil in the tank is decreasing. D. It tells what percent of oil remains after each hour. E. It gives the continuous hourly rate at which oil is leaking. F. It is the amount that the oil that leaks out each second. G. None of the above", "answer_v1": [ "100*(1-0.95^9)", "ln(0.95)", "CE" ], "answer_type_v1": [ "NV", "NV", "MCM" ], "options_v1": [ [], [], [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v2": "Oil leaks from a tank. At hour $t=0$ there are 210 gallons of oil in the tank. Each hour after that, 7\\% of the oil leaks out.\n(a) What percent of the original 210 gallons has leaked out after 6 hours? [ANS] \\% (b) If $\\ Q(t)=Q_0 e^{kt} \\ $ is the quantity of oil remaining after $t$ hours, find the value of $k$. $k=$ [ANS]\n(c) What does $k$ tell you about the leaking oil? Select all that apply if more than one statement is true [ANS] A. Because it is less than one, we know the amount of oil in the tank is decreasing. B. Because it is negative, we know the amount of oil in the tank is decreasing. C. It gives the continuous hourly rate at which oil is leaking. D. It tells what percent of oil remains after each hour. E. It tells by what percent of oil decays each hour. F. It is the amount that the oil that leaks out each second. G. None of the above", "answer_v2": [ "100*(1-0.93^6)", "ln(0.93)", "BC" ], "answer_type_v2": [ "NV", "NV", "MCM" ], "options_v2": [ [], [], [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v3": "Oil leaks from a tank. At hour $t=0$ there are 260 gallons of oil in the tank. Each hour after that, 6\\% of the oil leaks out.\n(a) What percent of the original 260 gallons has leaked out after 7 hours? [ANS] \\% (b) If $\\ Q(t)=Q_0 e^{kt} \\ $ is the quantity of oil remaining after $t$ hours, find the value of $k$. $k=$ [ANS]\n(c) What does $k$ tell you about the leaking oil? Select all that apply if more than one statement is true [ANS] A. It tells by what percent of oil decays each hour. B. It gives the continuous hourly rate at which oil is leaking. C. Because it is less than one, we know the amount of oil in the tank is decreasing. D. Because it is negative, we know the amount of oil in the tank is decreasing. E. It tells what percent of oil remains after each hour. F. It is the amount that the oil that leaks out each second. G. None of the above", "answer_v3": [ "100*(1-0.94^7)", "ln(0.94)", "BD" ], "answer_type_v3": [ "NV", "NV", "MCM" ], "options_v3": [ [], [], [ "A", "B", "C", "D", "E", "F", "G" ] ] }, { "id": "Algebra_0539", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "4", "keywords": [ "exponential functions", "continuous growth", "natural base", "e" ], "problem_v1": "World poultry production was 77.2 million tons in the year 2004 and increasing at a continuous rate of 1.6\\% per year. Assume that this growth rate continues.\n(a) Write an exponential formula for world poultry production, $P$, in million tons, as a function of the number of years, $t$, since 2004. $P=$ [ANS]\n(b) Use the formula to estimate world poultry production in the year 2015. [ANS] million tons (Round to the nearest 0.001.) (c) Use a graph to estimate the year in which world poultry production goes over 100 million tons. In the year [ANS]", "answer_v1": [ "77.2*exp(0.016*t)", "92.056", "2020" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "World poultry production was 77.2 million tons in the year 2004 and increasing at a continuous rate of 1.6\\% per year. Assume that this growth rate continues.\n(a) Write an exponential formula for world poultry production, $P$, in million tons, as a function of the number of years, $t$, since 2004. $P=$ [ANS]\n(b) Use the formula to estimate world poultry production in the year 2011. [ANS] million tons (Round to the nearest 0.001.) (c) Use a graph to estimate the year in which world poultry production goes over 105 million tons. In the year [ANS]", "answer_v2": [ "77.2*exp(0.016*t)", "86.349", "2023" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "World poultry production was 77.2 million tons in the year 2004 and increasing at a continuous rate of 1.6\\% per year. Assume that this growth rate continues.\n(a) Write an exponential formula for world poultry production, $P$, in million tons, as a function of the number of years, $t$, since 2004. $P=$ [ANS]\n(b) Use the formula to estimate world poultry production in the year 2012. [ANS] million tons (Round to the nearest 0.001.) (c) Use a graph to estimate the year in which world poultry production goes over 100 million tons. In the year [ANS]", "answer_v3": [ "77.2*exp(0.016*t)", "87.742", "2020" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0540", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "4", "keywords": [ "exponential functions", "annual growth rate", "linear growth" ], "problem_v1": "The number of asthma sufferers in the world was about 84 million in 1990 and 130 million in 2001. Let $N$ represent the number of asthma sufferers (in millions) worldwide $t$ years after 1990.\n(a) Write $N$ as a linear function of $t$. Use exact values in your formula, not decimal approximations. $N(t)=$ [ANS] million people (b) Write $N$ as an exponential function of $t$. Use exact values in your formula, not decimal approximations. $N(t)=$ [ANS] million people (c) How many asthma sufferers are predicted worldwide in the year 2018 with the linear model? [ANS] million people. (Round to the nearest 0.01 million people.) (d) How many asthma sufferers are predicted worldwide in the year 2018 with the exponential model? [ANS] million people. (Round to the nearest 0.01 million people.)", "answer_v1": [ "84+46/11*t", "84*(130/84)^(t/11)", "201.091", "255.306" ], "answer_type_v1": [ "EX", "EX", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "The number of asthma sufferers in the world was about 84 million in 1990 and 130 million in 2001. Let $N$ represent the number of asthma sufferers (in millions) worldwide $t$ years after 1990.\n(a) Write $N$ as a linear function of $t$. Use exact values in your formula, not decimal approximations. $N(t)=$ [ANS] million people (b) Write $N$ as an exponential function of $t$. Use exact values in your formula, not decimal approximations. $N(t)=$ [ANS] million people (c) How many asthma sufferers are predicted worldwide in the year 2010 with the linear model? [ANS] million people. (Round to the nearest 0.01 million people.) (d) How many asthma sufferers are predicted worldwide in the year 2010 with the exponential model? [ANS] million people. (Round to the nearest 0.01 million people.)", "answer_v2": [ "84+46/11*t", "84*(130/84)^(t/11)", "167.636", "185.833" ], "answer_type_v2": [ "EX", "EX", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "The number of asthma sufferers in the world was about 84 million in 1990 and 130 million in 2001. Let $N$ represent the number of asthma sufferers (in millions) worldwide $t$ years after 1990.\n(a) Write $N$ as a linear function of $t$. Use exact values in your formula, not decimal approximations. $N(t)=$ [ANS] million people (b) Write $N$ as an exponential function of $t$. Use exact values in your formula, not decimal approximations. $N(t)=$ [ANS] million people (c) How many asthma sufferers are predicted worldwide in the year 2013 with the linear model? [ANS] million people. (Round to the nearest 0.01 million people.) (d) How many asthma sufferers are predicted worldwide in the year 2013 with the exponential model? [ANS] million people. (Round to the nearest 0.01 million people.)", "answer_v3": [ "84+46/11*t", "84*(130/84)^(t/11)", "180.182", "209.339" ], "answer_type_v3": [ "EX", "EX", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Algebra_0541", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "4", "keywords": [ "exponential functions", "growth rate", "growth factor" ], "problem_v1": "What is the growth factor if water usage is increasing by 7\\% per year. Assume that time is measured in years. [ANS]", "answer_v1": [ "1.07" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "What is the growth factor if water usage is increasing by 3\\% per year. Assume that time is measured in years. [ANS]", "answer_v2": [ "1.03" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "What is the growth factor if water usage is increasing by 4\\% per year. Assume that time is measured in years. [ANS]", "answer_v3": [ "1.04" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0542", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "2", "keywords": [ "exponential functions", "growth rate", "growth factor" ], "problem_v1": "In the year 2004, a total of 7.6 million passengers took a cruise vacation. The global cruise industry has been growing at 9\\% per year for the last decade. Assume that this growth rate continues.\n(a) Write a formula for to approximate the number, $N$, of cruise passengers (in millions) $t$ years after 2004. $N=$ [ANS]\n(b) How many cruise passengers (in millions) are predicted in the year 2011? $N=$ [ANS]\n(c) How many cruise passengers (in millions) were there in the year 2000? $N=$ [ANS]", "answer_v1": [ "7.6*1.09^t", "13.8931", "5.38403" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "In the year 2001, a total of 7.2 million passengers took a cruise vacation. The global cruise industry has been growing at 8\\% per year for the last decade. Assume that this growth rate continues.\n(a) Write a formula for to approximate the number, $N$, of cruise passengers (in millions) $t$ years after 2001. $N=$ [ANS]\n(b) How many cruise passengers (in millions) are predicted in the year 2013? $N=$ [ANS]\n(c) How many cruise passengers (in millions) were there in the year 2000? $N=$ [ANS]", "answer_v2": [ "7.2*1.08^t", "18.1308", "6.66667" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "In the year 2002, a total of 7.3 million passengers took a cruise vacation. The global cruise industry has been growing at 9\\% per year for the last decade. Assume that this growth rate continues.\n(a) Write a formula for to approximate the number, $N$, of cruise passengers (in millions) $t$ years after 2002. $N=$ [ANS]\n(b) How many cruise passengers (in millions) are predicted in the year 2012? $N=$ [ANS]\n(c) How many cruise passengers (in millions) were there in the year 2000? $N=$ [ANS]", "answer_v3": [ "7.3*1.09^t", "17.2818", "6.14426" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0543", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "4", "keywords": [ "exponential functions", "growth rate", "growth factor" ], "problem_v1": "A typical cup of coffee contains about 100 mg of caffeine and every hour approximately 17\\% is metabolized and eliminated.\n(a) Write $C$, the amount of caffeine in the body in mg as a function of $t$, the number of hours since the coffee was consumed. $C(t)=$ [ANS] mg (b) How much caffeine is in the body after 5 hours? [ANS] mg (Round your answer to three decimal places.)", "answer_v1": [ "100*0.83^t", "39.39" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A typical cup of coffee contains about 120 mg of caffeine and every hour approximately 12\\% is metabolized and eliminated.\n(a) Write $C$, the amount of caffeine in the body in mg as a function of $t$, the number of hours since the coffee was consumed. $C(t)=$ [ANS] mg (b) How much caffeine is in the body after 3 hours? [ANS] mg (Round your answer to three decimal places.)", "answer_v2": [ "120*0.88^t", "81.777" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A typical cup of coffee contains about 110 mg of caffeine and every hour approximately 14\\% is metabolized and eliminated.\n(a) Write $C$, the amount of caffeine in the body in mg as a function of $t$, the number of hours since the coffee was consumed. $C(t)=$ [ANS] mg (b) How much caffeine is in the body after 4 hours? [ANS] mg (Round your answer to three decimal places.)", "answer_v3": [ "110*0.86^t", "60.171" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0544", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "4", "keywords": [ "exponential functions", "growth rate", "growth factor" ], "problem_v1": "The UN Food and Agriculture Organization estimates that $4.3$ \\% of the world's natural forests existing in 1990 were gone by the end of the decade. Suppose that in 1990, the world's forest cover stood at $3876$ million hectares. (The actual value was.)\n(a) Given the forest cover of $3876$ million hectares, how many millions of hectares of forest were lost during the 1990s? [ANS]\n(b) Based on your work in (a), how many million hectares of natural forests existed in the year 2000? [ANS]\n(c) Write an exponential formula approximating the number of million hectares of natural forest in the world $t$ years after 1990 if the value in 1990 is $3876$ million hectares. $P=f(t)=$ [ANS]\n(d) What was the annual percent decay rate during the 1990s? [ANS] \\% (Round to 0.01\\%)", "answer_v1": [ "166.668", "3709.33", "3876*0.957^(t/10)", "0.438554" ], "answer_type_v1": [ "NV", "NV", "EX", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "The UN Food and Agriculture Organization estimates that $4.5$ \\% of the world's natural forests existing in 1990 were gone by the end of the decade. Suppose that in 1990, the world's forest cover stood at $3808$ million hectares. (The actual value was.)\n(a) Given the forest cover of $3808$ million hectares, how many millions of hectares of forest were lost during the 1990s? [ANS]\n(b) Based on your work in (a), how many million hectares of natural forests existed in the year 2000? [ANS]\n(c) Write an exponential formula approximating the number of million hectares of natural forest in the world $t$ years after 1990 if the value in 1990 is $3808$ million hectares. $P=f(t)=$ [ANS]\n(d) What was the annual percent decay rate during the 1990s? [ANS] \\% (Round to 0.01\\%)", "answer_v2": [ "171.36", "3636.64", "3808*0.955^(t/10)", "0.459381" ], "answer_type_v2": [ "NV", "NV", "EX", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "The UN Food and Agriculture Organization estimates that $4.4$ \\% of the world's natural forests existing in 1990 were gone by the end of the decade. Suppose that in 1990, the world's forest cover stood at $3831$ million hectares. (The actual value was.)\n(a) Given the forest cover of $3831$ million hectares, how many millions of hectares of forest were lost during the 1990s? [ANS]\n(b) Based on your work in (a), how many million hectares of natural forests existed in the year 2000? [ANS]\n(c) Write an exponential formula approximating the number of million hectares of natural forest in the world $t$ years after 1990 if the value in 1990 is $3831$ million hectares. $P=f(t)=$ [ANS]\n(d) What was the annual percent decay rate during the 1990s? [ANS] \\% (Round to 0.01\\%)", "answer_v3": [ "168.564", "3662.44", "3831*0.956^(t/10)", "0.448963" ], "answer_type_v3": [ "NV", "NV", "EX", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Algebra_0545", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "4", "keywords": [ "exponential functions", "growth rate", "growth factor" ], "problem_v1": "Polluted water is passed through a series of filters. Each filter removes 82\\% of the remaining impurities. Initially, the untreated water contains impurities at a level of 420 parts per million (ppm). Find a formula for L, the remaining level of impurities, after the water has been passed through a series of n filters. $L(n)=$ [ANS] ppm", "answer_v1": [ "420*0.18^n" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Polluted water is passed through a series of filters. Each filter removes 76\\% of the remaining impurities. Initially, the untreated water contains impurities at a level of 440 parts per million (ppm). Find a formula for L, the remaining level of impurities, after the water has been passed through a series of n filters. $L(n)=$ [ANS] ppm", "answer_v2": [ "440*0.24^n" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Polluted water is passed through a series of filters. Each filter removes 78\\% of the remaining impurities. Initially, the untreated water contains impurities at a level of 430 parts per million (ppm). Find a formula for L, the remaining level of impurities, after the water has been passed through a series of n filters. $L(n)=$ [ANS] ppm", "answer_v3": [ "430*0.22^n" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0546", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "3", "keywords": [ "logarithms", "log", "ln", "half-life", "double-time", "continuous growth rate" ], "problem_v1": "In July 2005, the internet was linked by a global network of about 352.7 million host computers. The number of host computers has been growing approximately exponentially and was about 36.9 million in July 1998.\n(a) Find a formula for the number, N, N, of internet host computers (in millions of computers) as an exponential function of t, t, the number of years since July 1998, using the continuous exponential model $N(t)=ae^{kt}$. What are the values of a and k in your model? $a=$ [ANS]\n$k=$ [ANS] (Accurate to four decimal places)\n(b) Based on your equation above, what is the continuous annual percentage growth rate of N? Round your answer to the nearest 0.01\\%. By [ANS] \\%\n(c) What is the doubling time of N? [ANS] years (round your answer to the nearest 0.001 years)", "answer_v1": [ "36.9", "0.3225", "32.25", "2.149" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "In July 2005, the internet was linked by a global network of about 353.4 million host computers. The number of host computers has been growing approximately exponentially and was about 35.1 million in July 1998.\n(a) Find a formula for the number, N, N, of internet host computers (in millions of computers) as an exponential function of t, t, the number of years since July 1998, using the continuous exponential model $N(t)=ae^{kt}$. What are the values of a and k in your model? $a=$ [ANS]\n$k=$ [ANS] (Accurate to four decimal places)\n(b) Based on your equation above, what is the continuous annual percentage growth rate of N? Round your answer to the nearest 0.01\\%. By [ANS] \\%\n(c) What is the doubling time of N? [ANS] years (round your answer to the nearest 0.001 years)", "answer_v2": [ "35.1", "0.3299", "32.99", "2.101" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "In July 2005, the internet was linked by a global network of about 352.7 million host computers. The number of host computers has been growing approximately exponentially and was about 35.7 million in July 1998.\n(a) Find a formula for the number, N, N, of internet host computers (in millions of computers) as an exponential function of t, t, the number of years since July 1998, using the continuous exponential model $N(t)=ae^{kt}$. What are the values of a and k in your model? $a=$ [ANS]\n$k=$ [ANS] (Accurate to four decimal places)\n(b) Based on your equation above, what is the continuous annual percentage growth rate of N? Round your answer to the nearest 0.01\\%. By [ANS] \\%\n(c) What is the doubling time of N? [ANS] years (round your answer to the nearest 0.001 years)", "answer_v3": [ "35.7", "0.3272", "32.72", "2.118" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Algebra_0547", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "2", "keywords": [ "logarithms", "log", "ln", "half-life", "double-time", "continuous growth rate" ], "problem_v1": "The voltage V across a charged capacitor is given by $V(t)=6 e^{-0.5 t}$ where t is in seconds.\n(a) What is the voltage after 4 seconds? [ANS] volts (round to the nearest 0.001 volts)\n(b) When will the voltage be 1? In [ANS] seconds (round to the nearest 0.01 sec.)\n(c) By what percent does the voltage decrease each second? [ANS] \\% (round to the nearest 0.001\\%)", "answer_v1": [ "0.812", "3.58", "39.347" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The voltage V across a charged capacitor is given by $V(t)=3 e^{-0.3 t}$ where t is in seconds.\n(a) What is the voltage after 2 seconds? [ANS] volts (round to the nearest 0.001 volts)\n(b) When will the voltage be 1? In [ANS] seconds (round to the nearest 0.01 sec.)\n(c) By what percent does the voltage decrease each second? [ANS] \\% (round to the nearest 0.001\\%)", "answer_v2": [ "1.646", "3.66", "25.918" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The voltage V across a charged capacitor is given by $V(t)=4 e^{-0.4 t}$ where t is in seconds.\n(a) What is the voltage after 3 seconds? [ANS] volts (round to the nearest 0.001 volts)\n(b) When will the voltage be 1? In [ANS] seconds (round to the nearest 0.01 sec.)\n(c) By what percent does the voltage decrease each second? [ANS] \\% (round to the nearest 0.001\\%)", "answer_v3": [ "1.205", "3.47", "32.968" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0548", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "5", "keywords": [ "decomposition", "inverse", "composition", "combinations", "function" ], "problem_v1": "A hot brick is removed from a kiln at $250^{\\circ}$ C above room temperature. Over time, the brick cools off. After 2 hours have elapsed, the brick is $20^{\\circ}$ C above room temperature. Let $t$ be the time in hours since the brick was removed from the kiln. Let $y=H(t)$ be the difference between the brick's and the room's temperature at time $t$. Assume that $H(t)$ is an exponential function.\n(a) Find a formula for $H(t)$. $H(t)=$ [ANS]. (b) How many degrees does the brick's temperature drop during the first quarter hour? It drops [ANS] $^{\\circ}$ C. (c) How many degrees does the temperature drop during the next quarter hour? It drops [ANS] $^{\\circ}$ C. (d) Find $H^{-1}(y)$. $H^{-1}(y)=$ [ANS]. (e) How much time elapses before the brick's temperature is $5^{\\circ}$ C above room temperature? time=[ANS] hours.", "answer_v1": [ "250*e^(-[ln(250/20)]/2*t)", "250-250*e^(-[ln(250/20)]/2*0.25)", "250*e^(-[ln(250/20)]/2*0.25)-250*e^(-[ln(250/20)]/2*0.5)", "-[ln(y/250)]/([ln(250/20)]/2)", "-[ln(5/250)]/([ln(250/20)]/2)" ], "answer_type_v1": [ "EX", "NV", "NV", "EX", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "A hot brick is removed from a kiln at $100^{\\circ}$ C above room temperature. Over time, the brick cools off. After 2 hours have elapsed, the brick is $30^{\\circ}$ C above room temperature. Let $t$ be the time in hours since the brick was removed from the kiln. Let $y=H(t)$ be the difference between the brick's and the room's temperature at time $t$. Assume that $H(t)$ is an exponential function.\n(a) Find a formula for $H(t)$. $H(t)=$ [ANS]. (b) How many degrees does the brick's temperature drop during the first quarter hour? It drops [ANS] $^{\\circ}$ C. (c) How many degrees does the temperature drop during the next quarter hour? It drops [ANS] $^{\\circ}$ C. (d) Find $H^{-1}(y)$. $H^{-1}(y)=$ [ANS]. (e) How much time elapses before the brick's temperature is $3^{\\circ}$ C above room temperature? time=[ANS] hours.", "answer_v2": [ "100*e^(-[ln(100/30)]/2*t)", "100-100*e^(-[ln(100/30)]/2*0.25)", "100*e^(-[ln(100/30)]/2*0.25)-100*e^(-[ln(100/30)]/2*0.5)", "-[ln(y/100)]/([ln(100/30)]/2)", "-[ln(3/100)]/([ln(100/30)]/2)" ], "answer_type_v2": [ "EX", "NV", "NV", "EX", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "A hot brick is removed from a kiln at $150^{\\circ}$ C above room temperature. Over time, the brick cools off. After 2 hours have elapsed, the brick is $25^{\\circ}$ C above room temperature. Let $t$ be the time in hours since the brick was removed from the kiln. Let $y=H(t)$ be the difference between the brick's and the room's temperature at time $t$. Assume that $H(t)$ is an exponential function.\n(a) Find a formula for $H(t)$. $H(t)=$ [ANS]. (b) How many degrees does the brick's temperature drop during the first quarter hour? It drops [ANS] $^{\\circ}$ C. (c) How many degrees does the temperature drop during the next quarter hour? It drops [ANS] $^{\\circ}$ C. (d) Find $H^{-1}(y)$. $H^{-1}(y)=$ [ANS]. (e) How much time elapses before the brick's temperature is $4^{\\circ}$ C above room temperature? time=[ANS] hours.", "answer_v3": [ "150*e^(-[ln(150/25)]/2*t)", "150-150*e^(-[ln(150/25)]/2*0.25)", "150*e^(-[ln(150/25)]/2*0.25)-150*e^(-[ln(150/25)]/2*0.5)", "-[ln(y/150)]/([ln(150/25)]/2)", "-[ln(4/150)]/([ln(150/25)]/2)" ], "answer_type_v3": [ "EX", "NV", "NV", "EX", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Algebra_0549", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "4", "keywords": [ "Algebra", "Exponential", "Logarithmic", "Applications", "exponential equation", "modeling" ], "problem_v1": "A roasted turkey is taken from an oven when its temperature has reached 185 Fahrenheit and is placed on a table in a room where the temperature is 75 Fahrenheit.\n(a) If the temperature of the turkey is 155 Fahrenheit after half an hour, what is its temperature after 45 minutes? Your answer is [ANS] Fahrenheit. (b) When will the trukey cool to 100 Fahrenheit? Your answer is [ANS] hours.", "answer_v1": [ "143.224229233795", "2.32624773420025" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A roasted turkey is taken from an oven when its temperature has reached 185 Fahrenheit and is placed on a table in a room where the temperature is 75 Fahrenheit.\n(a) If the temperature of the turkey is 141 Fahrenheit after half an hour, what is its temperature after 45 minutes? Your answer is [ANS] Fahrenheit. (b) When will the trukey cool to 100 Fahrenheit? Your answer is [ANS] hours.", "answer_v2": [ "126.123380169938", "1.45020577668098" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A roasted turkey is taken from an oven when its temperature has reached 185 Fahrenheit and is placed on a table in a room where the temperature is 75 Fahrenheit.\n(a) If the temperature of the turkey is 146 Fahrenheit after half an hour, what is its temperature after 45 minutes? Your answer is [ANS] Fahrenheit. (b) When will the trukey cool to 100 Fahrenheit? Your answer is [ANS] hours.", "answer_v3": [ "132.041531918738", "1.69210014492074" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0550", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "4", "keywords": [ "Algebra", "Exponential", "Logarithmic", "Applications", "exponential equation", "modeling" ], "problem_v1": "An infectious strain of bacteria increases in number at a relative growth rate of 250 percent per hour. When a certain critical number of bacteria are present in the bloodstream, a person becomes ill. If a single bacterium infects a person, the critical level is reached in 24 hours. How long will it take for the critical level to be reached if the same person is infected with 10 bacteria? Your answer is [ANS] hours.", "answer_v1": [ "23.0789659628024" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "An infectious strain of bacteria increases in number at a relative growth rate of 110 percent per hour. When a certain critical number of bacteria are present in the bloodstream, a person becomes ill. If a single bacterium infects a person, the critical level is reached in 24 hours. How long will it take for the critical level to be reached if the same person is infected with 10 bacteria? Your answer is [ANS] hours.", "answer_v2": [ "21.9067408245509" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "An infectious strain of bacteria increases in number at a relative growth rate of 160 percent per hour. When a certain critical number of bacteria are present in the bloodstream, a person becomes ill. If a single bacterium infects a person, the critical level is reached in 24 hours. How long will it take for the critical level to be reached if the same person is infected with 10 bacteria? Your answer is [ANS] hours.", "answer_v3": [ "22.5608843168787" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0551", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "4", "keywords": [ "Algebra", "Exponential", "Logarithmic", "Applications", "logarithms" ], "problem_v1": "The 1906 San Francisco earthquake had a magnitude of 8.3 on the Richter scale. At the same time in South America there was an eathquake with magnitude 5.1 that caused only minor damage. How many times more intense was the San Francisco earthquake than the South American one? [ANS]", "answer_v1": [ "1584.89319246112" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The 1906 San Francisco earthquake had a magnitude of 8.3 on the Richter scale. At the same time in South America there was an eathquake with magnitude 4 that caused only minor damage. How many times more intense was the San Francisco earthquake than the South American one? [ANS]", "answer_v2": [ "19952.6231496888" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The 1906 San Francisco earthquake had a magnitude of 8.3 on the Richter scale. At the same time in South America there was an eathquake with magnitude 4.4 that caused only minor damage. How many times more intense was the San Francisco earthquake than the South American one? [ANS]", "answer_v3": [ "7943.28234724282" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0552", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "4", "keywords": [ "Algebra", "Exponential", "Logarithmic", "Applications", "solve for variable' 'fraction" ], "problem_v1": "Let $P(t)=33 (1-e^{-kt})+62$ represent the expected score for a student who studies $t$ hours for a test. Suppose $k=0.35$ and test scores must be integers.\nWhat is the highest score the student can expect? [ANS]\nIf the student does not study, what score can he expect? [ANS]", "answer_v1": [ "95", "62" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $P(t)=48 (1-e^{-kt})+51$ represent the expected score for a student who studies $t$ hours for a test. Suppose $k=0.16$ and test scores must be integers.\nWhat is the highest score the student can expect? [ANS]\nIf the student does not study, what score can he expect? [ANS]", "answer_v2": [ "99", "51" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $P(t)=37 (1-e^{-kt})+55$ represent the expected score for a student who studies $t$ hours for a test. Suppose $k=0.21$ and test scores must be integers.\nWhat is the highest score the student can expect? [ANS]\nIf the student does not study, what score can he expect? [ANS]", "answer_v3": [ "92", "55" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0553", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "4", "keywords": [ "algebra", "solve for variable' 'fraction", "Exponential", "Logarithmic", "Applications" ], "problem_v1": "The town of Sickville, with a population of 2474 is exposed to the Blue Moon Virus, against which there is no immunity. The number of people infected when the virus is detected is 65. Suppose the number of infections grows logistically, with $k=0.53$.\nFind $A$. [ANS]\nFind the formula for the number of people infected after $t$ days. $N(t)=$ [ANS]\nFind the number of people infected after 25 days. [ANS]", "answer_v1": [ "37.0615384615385", "2474/(1 + 37.0615384615385*2.71828182845905**(-0.53*t))", "2473.83860394299" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The town of Sickville, with a population of 9170 is exposed to the Blue Moon Virus, against which there is no immunity. The number of people infected when the virus is detected is 95. Suppose the number of infections grows logistically, with $k=0.13$.\nFind $A$. [ANS]\nFind the formula for the number of people infected after $t$ days. $N(t)=$ [ANS]\nFind the number of people infected after 17 days. [ANS]", "answer_v2": [ "95.5263157894737", "9170/(1 + 95.5263157894737*2.71828182845905**(-0.13*t))", "798.82928093734" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The town of Sickville, with a population of 6866 is exposed to the Blue Moon Virus, against which there is no immunity. The number of people infected when the virus is detected is 65. Suppose the number of infections grows logistically, with $k=0.24$.\nFind $A$. [ANS]\nFind the formula for the number of people infected after $t$ days. $N(t)=$ [ANS]\nFind the number of people infected after 21 days. [ANS]", "answer_v3": [ "104.630769230769", "6866/(1 + 104.630769230769*2.71828182845905**(-0.24*t))", "4093.35358135067" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0554", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "3", "keywords": [ "logarithms", "exponentials", "exponential growth", "decay", "Algebra", "Exponential", "Logarithmic", "Applications" ], "problem_v1": "You are taking a road trip in a car without A/C. The temperture in the car is 104 degrees F. You buy a cold pop at a gas station. Its initial temperature is 45 degrees F. The pop's temperature reaches 60 degrees F after 32 minutes. Given that \\frac{T-A}{T_0-A}=e^{-kt} where $T=$ the temperature of the pop at time t. $T_0=$ the initial temperature of the pop. $A=$ the temperature in the car. $k=$ a constant that corresponds to the warming rate. and $t=$ the length of time that the pop has been warming up.\nHow long will it take the pop to reach a temperature of 78.75 degrees F? It will take [ANS] minutes.", "answer_v1": [ "92.5821168498253" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "You are taking a road trip in a car without A/C. The temperture in the car is 87 degrees F. You buy a cold pop at a gas station. Its initial temperature is 45 degrees F. The pop's temperature reaches 60 degrees F after 45 minutes. Given that \\frac{T-A}{T_0-A}=e^{-kt} where $T=$ the temperature of the pop at time t. $T_0=$ the initial temperature of the pop. $A=$ the temperature in the car. $k=$ a constant that corresponds to the warming rate. and $t=$ the length of time that the pop has been warming up.\nHow long will it take the pop to reach a temperature of 72 degrees F? It will take [ANS] minutes.", "answer_v2": [ "104.865186055493" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "You are taking a road trip in a car without A/C. The temperture in the car is 93 degrees F. You buy a cold pop at a gas station. Its initial temperature is 45 degrees F. The pop's temperature reaches 60 degrees F after 33 minutes. Given that \\frac{T-A}{T_0-A}=e^{-kt} where $T=$ the temperature of the pop at time t. $T_0=$ the initial temperature of the pop. $A=$ the temperature in the car. $k=$ a constant that corresponds to the warming rate. and $t=$ the length of time that the pop has been warming up.\nHow long will it take the pop to reach a temperature of 73.75 degrees F? It will take [ANS] minutes.", "answer_v3": [ "80.4704976846945" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0555", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "5", "keywords": [ "calculus", "logarithmic functions", "logarithms", "laws of logarithms" ], "problem_v1": "Suppose that the sales at Borders bookstores went from $76$ million dollars in $1991$ to $419$ million dollars in $1995$. Find an exponential function to model the sales (in millions of dollars) as a function of years, $t$, since $1991$. sales=[ANS] million dollars What is the continuous percent growth rate, per year, of sales? continuous growth rate=[ANS] percent\n(Enter your answer as a percent, not a fraction.)", "answer_v1": [ "76*e^([ln(419/76)]/4*t)", "100*[ln(419/76)]/4" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose that the sales at Borders bookstores went from $60$ million dollars in $1993$ to $404$ million dollars in $1996$. Find an exponential function to model the sales (in millions of dollars) as a function of years, $t$, since $1993$. sales=[ANS] million dollars What is the continuous percent growth rate, per year, of sales? continuous growth rate=[ANS] percent\n(Enter your answer as a percent, not a fraction.)", "answer_v2": [ "60*e^([ln(404/60)]/3*t)", "100*[ln(404/60)]/3" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose that the sales at Borders bookstores went from $66$ million dollars in $1992$ to $408$ million dollars in $1996$. Find an exponential function to model the sales (in millions of dollars) as a function of years, $t$, since $1992$. sales=[ANS] million dollars What is the continuous percent growth rate, per year, of sales? continuous growth rate=[ANS] percent\n(Enter your answer as a percent, not a fraction.)", "answer_v3": [ "66*e^([ln(408/66)]/4*t)", "100*[ln(408/66)]/4" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0556", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "5", "keywords": [ "algebra" ], "problem_v1": "A sound intensity $I$ is measured by the decibel scale \\beta=10 \\log_{10}\\frac{I}{I_0} where $I_0$ is a reference intensity. Thus adding 10 decibels to $\\beta$ multiplies $I$ with 10. Adding 7 decibels multiplies $I$ with [ANS].", "answer_v1": [ "5.01187233627272" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A sound intensity $I$ is measured by the decibel scale \\beta=10 \\log_{10}\\frac{I}{I_0} where $I_0$ is a reference intensity. Thus adding 10 decibels to $\\beta$ multiplies $I$ with 10. Adding 2 decibels multiplies $I$ with [ANS].", "answer_v2": [ "1.58489319246111" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A sound intensity $I$ is measured by the decibel scale \\beta=10 \\log_{10}\\frac{I}{I_0} where $I_0$ is a reference intensity. Thus adding 10 decibels to $\\beta$ multiplies $I$ with 10. Adding 4 decibels multiplies $I$ with [ANS].", "answer_v3": [ "2.51188643150958" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0557", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "4", "keywords": [ "algebra" ], "problem_v1": "The logistic curve is the graph of the function f(x)=\\frac{a}{1+be^{-rx}} where $a$, $b$, and $r$ are suitable parameters. This function may describe, for example, the initial rapid growth of a population, followed by a slowdown of the growth as resources become sparse. Since $e^{-rx}$ tends to zero as $x$ tends to infinity $f(x)$ approaches $a$ as $x$ tends to infinity, and at $x=0$ the initial population equals f(0)=\\frac{a}{1+b}. The rate $r$ is the usual growth rate that would prevail indefinitely in the presence of unlimited resources.\nSuppose $a=3800$, $b=60$, and $r=0.01$. Then the initial population is [ANS] and as time goes on the population approaches but never quite reaches [ANS].", "answer_v1": [ "60", "3800" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The logistic curve is the graph of the function f(x)=\\frac{a}{1+be^{-rx}} where $a$, $b$, and $r$ are suitable parameters. This function may describe, for example, the initial rapid growth of a population, followed by a slowdown of the growth as resources become sparse. Since $e^{-rx}$ tends to zero as $x$ tends to infinity $f(x)$ approaches $a$ as $x$ tends to infinity, and at $x=0$ the initial population equals f(0)=\\frac{a}{1+b}. The rate $r$ is the usual growth rate that would prevail indefinitely in the presence of unlimited resources.\nSuppose $a=500$, $b=90$, and $r=0.01$. Then the initial population is [ANS] and as time goes on the population approaches but never quite reaches [ANS].", "answer_v2": [ "90", "500" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The logistic curve is the graph of the function f(x)=\\frac{a}{1+be^{-rx}} where $a$, $b$, and $r$ are suitable parameters. This function may describe, for example, the initial rapid growth of a population, followed by a slowdown of the growth as resources become sparse. Since $e^{-rx}$ tends to zero as $x$ tends to infinity $f(x)$ approaches $a$ as $x$ tends to infinity, and at $x=0$ the initial population equals f(0)=\\frac{a}{1+b}. The rate $r$ is the usual growth rate that would prevail indefinitely in the presence of unlimited resources.\nSuppose $a=1600$, $b=60$, and $r=0.01$. Then the initial population is [ANS] and as time goes on the population approaches but never quite reaches [ANS].", "answer_v3": [ "60", "1600" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0558", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "5", "keywords": [ "exponents" ], "problem_v1": "The area of a wetland drops by $\\frac{1}{6}$ every $7$ years. What percent of its total area disappears after $35$ years?\nPercent lost $=$ [ANS] \\%", "answer_v1": [ "59.8122" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The area of a wetland drops by $\\frac{1}{3}$ every $9$ years. What percent of its total area disappears after $45$ years?\nPercent lost $=$ [ANS] \\%", "answer_v2": [ "86.8313" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The area of a wetland drops by $\\frac{1}{4}$ every $7$ years. What percent of its total area disappears after $35$ years?\nPercent lost $=$ [ANS] \\%", "answer_v3": [ "76.2695" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0559", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "2", "keywords": [ "exponential functions", "growth rate", "growth factor" ], "problem_v1": "The area covered by a marsh, $A$, starts at $175$ acres and drops by a factor of $\\dfrac{1}{4}$ each year for $n$ years. Write a formula for $A=f(n)$.\n$A=f(n)=$ [ANS]", "answer_v1": [ "175*0.75^n" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "The area covered by a marsh, $A$, starts at $105$ acres and drops by a factor of $\\dfrac{1}{5}$ each year for $n$ years. Write a formula for $A=f(n)$.\n$A=f(n)=$ [ANS]", "answer_v2": [ "105*0.8^n" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "The area covered by a marsh, $A$, starts at $130$ acres and drops by a factor of $\\dfrac{1}{4}$ each year for $n$ years. Write a formula for $A=f(n)$.\n$A=f(n)=$ [ANS]", "answer_v3": [ "130*0.75^n" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0560", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "5", "keywords": [ "exponential functions", "annual growth rate", "linear growth" ], "problem_v1": "Nicotine leaves the body at a constant rate. At a time of $4$ hours after smoking a cigarette, $60$ mg of nicotine remain in a person's body; $7$ hours later ($11$ hours after smoking the cigarette), there are $15$ mg of nicotine in the body.\n(a) Find a formula for the amount of nicotine $A$ left in the body $t$ hours after smoking a cigarette.\n$A=f(t)=$ [ANS]\n(b) How much nicotine is in the body immediately after smoking. [ANS]\n(c) How much nicotine is in the body $6$ hours after smoking. [ANS]", "answer_v1": [ "132.491*0.25^(t/7)", "132.491", "40.377" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Nicotine leaves the body at a constant rate. At a time of $2$ hours after smoking a cigarette, $52$ mg of nicotine remain in a person's body; $9$ hours later ($11$ hours after smoking the cigarette), there are $13$ mg of nicotine in the body.\n(a) Find a formula for the amount of nicotine $A$ left in the body $t$ hours after smoking a cigarette.\n$A=f(t)=$ [ANS]\n(b) How much nicotine is in the body immediately after smoking. [ANS]\n(c) How much nicotine is in the body $6$ hours after smoking. [ANS]", "answer_v2": [ "70.7611*0.25^(t/9)", "70.7611", "28.0816" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Nicotine leaves the body at a constant rate. At a time of $2$ hours after smoking a cigarette, $56$ mg of nicotine remain in a person's body; $8$ hours later ($10$ hours after smoking the cigarette), there are $14$ mg of nicotine in the body.\n(a) Find a formula for the amount of nicotine $A$ left in the body $t$ hours after smoking a cigarette.\n$A=f(t)=$ [ANS]\n(b) How much nicotine is in the body immediately after smoking. [ANS]\n(c) How much nicotine is in the body $6$ hours after smoking. [ANS]", "answer_v3": [ "79.196*0.25^(t/8)", "79.196", "28" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0561", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "5", "keywords": [ "Algebra", "logarithmic" ], "problem_v1": "The velocity (in ft/s) of a sky diver $t$ seconds after jumping is given by v(t)=85 (1-e^{-0.2 t}) After how many seconds is the velocity 75 ft/s? [ANS] seconds", "answer_v1": [ "-ln(1-75/85)/0.2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The velocity (in ft/s) of a sky diver $t$ seconds after jumping is given by v(t)=60 (1-e^{-0.2 t}) After how many seconds is the velocity 55 ft/s? [ANS] seconds", "answer_v2": [ "-ln(1-55/60)/0.2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The velocity (in ft/s) of a sky diver $t$ seconds after jumping is given by v(t)=70 (1-e^{-0.2 t}) After how many seconds is the velocity 65 ft/s? [ANS] seconds", "answer_v3": [ "-ln(1-65/70)/0.2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0562", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "5", "keywords": [ "Algebra", "logarithmic" ], "problem_v1": "For a pole-vaulter in training, the height the athelete will be able to pole-vault after $t$ months of training is given by H(t)=20-13 e^{-0.02 t} After how many months will she be able to vault 14 ft? [ANS] months", "answer_v1": [ "-ln((20-14)/13)/0.02" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "For a pole-vaulter in training, the height the athelete will be able to pole-vault after $t$ months of training is given by H(t)=17-8 e^{-0.02 t} After how many months will she be able to vault 12 ft? [ANS] months", "answer_v2": [ "-ln((17-12)/8)/0.02" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "For a pole-vaulter in training, the height the athelete will be able to pole-vault after $t$ months of training is given by H(t)=18-10 e^{-0.02 t} After how many months will she be able to vault 12 ft? [ANS] months", "answer_v3": [ "-ln((18-12)/10)/0.02" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0563", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "4", "keywords": [ "exponential model", "growth" ], "problem_v1": "A country's consumption of iron increases 5.7\\% per year. Assuming this rate of increase, its demand during the next year will be 8,000,000 tons. The country has 158 years of iron reserves at this rate of consumption.\n(a) How much iron will be used at the end of 13 years? Answer: [ANS] tons (b) How long will the iron reserves last? Answer: [ANS] years (c) If residents of the country can reduce the growth rate of their consumption by 1.1\\% per year, how long will the iron reserves last? Answer: [ANS] years", "answer_v1": [ "148178452.176242", "41.5477061638511", "46.9698589738733" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A country's consumption of coal increases 2.9\\% per year. Assuming this rate of increase, its demand during the next year will be 4,000,000 tons. The country has 194 years of coal reserves at this rate of consumption.\n(a) How much coal will be used at the end of 21 years? Answer: [ANS] tons (b) How long will the coal reserves last? Answer: [ANS] years (c) If residents of the country can reduce the growth rate of their consumption by 1.5\\% per year, how long will the coal reserves last? Answer: [ANS] years", "answer_v2": [ "113480305.886391", "66.1479373164738", "94.4153617532598" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A country's consumption of oil increases 3.6\\% per year. Assuming this rate of increase, its demand during the next year will be 6,000,000 barrels. The country has 161 years of oil reserves at this rate of consumption.\n(a) How much oil will be used at the end of 18 years? Answer: [ANS] barrels (b) How long will the oil reserves last? Answer: [ANS] years (c) If residents of the country can reduce the growth rate of their consumption by 1.1\\% per year, how long will the oil reserves last? Answer: [ANS] years", "answer_v3": [ "148343337.892108", "54.1840249420732", "65.3809088866791" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0564", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "4", "keywords": [ "word problem" ], "problem_v1": "The probability that a \\$1 ticket DOES NOT win in the State Lottery is $\\frac{7258072}{7258073}$. The probabilty $p$ that $n$ independently sold tickets are ALL losers and the jackpot rolls over is given by $p=(\\frac{7258072}{7258073})^n$. As $n$ increases, is $p$ increasing or decreasing? In other words, as the number of tickets sold increases, is the probability of a rollover increasing or decreasing? [ANS]\nFor what number of tickets sold is the probability of a rollover greater than 55\\%? Fewer than [ANS] tickets must be sold.", "answer_v1": [ "DECREASING", "4339144.29375749" ], "answer_type_v1": [ "MCS", "NV" ], "options_v1": [ [ "increasing", "decreasing" ], [] ], "problem_v2": "The probability that a \\$1 ticket DOES NOT win in the State Lottery is $\\frac{5249052}{5249053}$. The probabilty $p$ that $n$ independently sold tickets are ALL losers and the jackpot rolls over is given by $p=(\\frac{5249052}{5249053})^n$. As $n$ increases, is $p$ increasing or decreasing? In other words, as the number of tickets sold increases, is the probability of a rollover increasing or decreasing? [ANS]\nFor what number of tickets sold is the probability of a rollover greater than 75\\%? Fewer than [ANS] tickets must be sold.", "answer_v2": [ "DECREASING", "1510058.30123835" ], "answer_type_v2": [ "MCS", "NV" ], "options_v2": [ [ "increasing", "decreasing" ], [] ], "problem_v3": "The probability that a \\$1 ticket DOES NOT win in the State Lottery is $\\frac{5940347}{5940348}$. The probabilty $p$ that $n$ independently sold tickets are ALL losers and the jackpot rolls over is given by $p=(\\frac{5940347}{5940348})^n$. As $n$ increases, is $p$ increasing or decreasing? In other words, as the number of tickets sold increases, is the probability of a rollover increasing or decreasing? [ANS]\nFor what number of tickets sold is the probability of a rollover greater than 55\\%? Fewer than [ANS] tickets must be sold.", "answer_v3": [ "DECREASING", "3551359.53201527" ], "answer_type_v3": [ "MCS", "NV" ], "options_v3": [ [ "increasing", "decreasing" ], [] ] }, { "id": "Algebra_0565", "subject": "Algebra", "topic": "Exponential and logarithmic expressions and functions", "subtopic": "Applications and models - general", "level": "4", "keywords": [ "word problem" ], "problem_v1": "The cost of a parking ticket at NAU is \\$40 for the first offense, but the cost triples for each additional offense. Write a formula for the cost $C$ as a function of the number of tickets $n$. Remember to use the variable \" $n$ \" in your answer. $\\ C$=[ANS]", "answer_v1": [ "40*3**(n-1)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "The cost of a parking ticket at NAU is \\$10 for the first offense, but the cost triples for each additional offense. Write a formula for the cost $C$ as a function of the number of tickets $n$. Remember to use the variable \" $n$ \" in your answer. $\\ C$=[ANS]", "answer_v2": [ "10*3**(n-1)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "The cost of a parking ticket at NAU is \\$20 for the first offense, but the cost triples for each additional offense. Write a formula for the cost $C$ as a function of the number of tickets $n$. Remember to use the variable \" $n$ \" in your answer. $\\ C$=[ANS]", "answer_v3": [ "20*3**(n-1)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0566", "subject": "Algebra", "topic": "Finite sequences and series", "subtopic": "Notation", "level": "3", "keywords": [], "problem_v1": "Write the sum in summation notation, using the given lower bound.\n6+12+18+24+30+36=\\sum_{i=1}^{b} a_i $b=$ [ANS], $a_i=$ [ANS]", "answer_v1": [ "6", "6*i" ], "answer_type_v1": [ "NV", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Write the sum in summation notation, using the given lower bound.\n3+6+9+12+15+18+21+24=\\sum_{i=1}^{b} a_i $b=$ [ANS], $a_i=$ [ANS]", "answer_v2": [ "8", "3*i" ], "answer_type_v2": [ "NV", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Write the sum in summation notation, using the given lower bound.\n4+8+12+16+20+24+28=\\sum_{i=1}^{b} a_i $b=$ [ANS], $a_i=$ [ANS]", "answer_v3": [ "7", "4*i" ], "answer_type_v3": [ "NV", "EX" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0567", "subject": "Algebra", "topic": "Finite sequences and series", "subtopic": "Notation", "level": "2", "keywords": [ "calculus", "series" ], "problem_v1": "Find a formula for the $n$ th term of the following sequences:\n(a) $\\frac{1}{1},\\frac{-1}{16},\\frac{1}{81},\\ldots$ $a_n=$ [ANS]\n(b) $\\frac{3}{5},\\frac{4}{6},\\frac{5}{7},\\ldots$ $b_n=$ [ANS]\nAssume a starting index of $n=1$.", "answer_v1": [ "(-1)^(n+1)/(n^4)", "(n+2)/(n+4)" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Find a formula for the $n$ th term of the following sequences:\n(a) $\\frac{1}{1},\\frac{-1}{4},\\frac{1}{9},\\ldots$ $a_n=$ [ANS]\n(b) $\\frac{4}{5},\\frac{5}{6},\\frac{6}{7},\\ldots$ $b_n=$ [ANS]\nAssume a starting index of $n=1$.", "answer_v2": [ "(-1)^(n+1)/(n^2)", "(n+3)/(n+4)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Find a formula for the $n$ th term of the following sequences:\n(a) $\\frac{1}{1},\\frac{-1}{4},\\frac{1}{9},\\ldots$ $a_n=$ [ANS]\n(b) $\\frac{3}{4},\\frac{4}{5},\\frac{5}{6},\\ldots$ $b_n=$ [ANS]\nAssume a starting index of $n=1$.", "answer_v3": [ "(-1)^(n+1)/(n^2)", "(n+2)/(n+3)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0568", "subject": "Algebra", "topic": "Finite sequences and series", "subtopic": "Notation", "level": "1", "keywords": [ "Sum", "Sigma Notation" ], "problem_v1": "Find the numerical value of the sum below. $ \\sum\\limits_{k=1}^{120} 17$=[ANS]", "answer_v1": [ "2040" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the numerical value of the sum below. $ \\sum\\limits_{k=1}^{80} 24$=[ANS]", "answer_v2": [ "1920" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the numerical value of the sum below. $ \\sum\\limits_{k=1}^{95} 17$=[ANS]", "answer_v3": [ "1615" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0569", "subject": "Algebra", "topic": "Finite sequences and series", "subtopic": "Notation", "level": "2", "keywords": [ "Sequences" ], "problem_v1": "$ \\sum_{i=1}^{12} [1+(-1)^i]=$ [ANS].", "answer_v1": [ "12" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "$ \\sum_{i=1}^{6} [1+(-1)^i]=$ [ANS].", "answer_v2": [ "6" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "$ \\sum_{i=1}^{8} [1+(-1)^i]=$ [ANS].", "answer_v3": [ "8" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0570", "subject": "Algebra", "topic": "Finite sequences and series", "subtopic": "Notation", "level": "3", "keywords": [ "calculus", "integral", "sequences" ], "problem_v1": "In electrical engineering, a continuous function like $f(t)=\\sin t$, where $t$ is in seconds, is referred to as an analog signal. To digitize the signal, we sample $f(t)$ every $\\Delta t$ seconds to form the sequence $s_n=f(n\\Delta t)$. For example, sampling $f$ every 1/10 second produces the sequence $\\sin(1/10)$, $\\sin(2/10)$, $\\sin(3/10)$,... Suppose that the analog signal is given by f(t)={\\sin(1.5 t)\\over t}. Give the first 6 terms of a sampling of the signal every $\\Delta t=1.5$ seconds: [ANS]\n(Enter your answer as a comma-separated list.)", "answer_v1": [ "(0.518715, -0.325843, 0.10001, 0.0686864, -0.129041, 0.0893094)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "In electrical engineering, a continuous function like $f(t)=\\sin t$, where $t$ is in seconds, is referred to as an analog signal. To digitize the signal, we sample $f(t)$ every $\\Delta t$ seconds to form the sequence $s_n=f(n\\Delta t)$. For example, sampling $f$ every 1/10 second produces the sequence $\\sin(1/10)$, $\\sin(2/10)$, $\\sin(3/10)$,... Suppose that the analog signal is given by f(t)={\\sin(0.5 t)\\over t}. Give the first 6 terms of a sampling of the signal every $\\Delta t=0.5$ seconds: [ANS]\n(Enter your answer as a comma-separated list.)", "answer_v2": [ "(0.494808, 0.479426, 0.454426, 0.420735, 0.379594, 0.332498)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "In electrical engineering, a continuous function like $f(t)=\\sin t$, where $t$ is in seconds, is referred to as an analog signal. To digitize the signal, we sample $f(t)$ every $\\Delta t$ seconds to form the sequence $s_n=f(n\\Delta t)$. For example, sampling $f$ every 1/10 second produces the sequence $\\sin(1/10)$, $\\sin(2/10)$, $\\sin(3/10)$,... Suppose that the analog signal is given by f(t)={\\sin(0.5 t)\\over t}. Give the first 6 terms of a sampling of the signal every $\\Delta t=1$ seconds: [ANS]\n(Enter your answer as a comma-separated list.)", "answer_v3": [ "(0.479426, 0.420735, 0.332498, 0.227324, 0.119694, 0.02352)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Algebra_0571", "subject": "Algebra", "topic": "Finite sequences and series", "subtopic": "Notation", "level": "2", "keywords": [ "calculus", "integral", "sequences" ], "problem_v1": "World oil consumption can be estimated to be approximately 75.974 million barrels per day in 2003 and to be increasing at approximately 0.3 percent per year. Let $c_n$ be the daily world oil consumption (in millions of barrels per day) $n$ years after 2003. A. Find a formula for $c_n$: $c_n$=[ANS]\nB. Find $c_n-c_{n-1}$: $c_n-c_{n-1}$=[ANS]\n(Be sure that you can interpret what this difference means in practical terms.) (Be sure that you can interpret what this difference means in practical terms.)", "answer_v1": [ "75.974*(1 + 0.3/100)^n", "75.974*((1 + 0.3/100)^(n-1))*(0.3/100)" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "World oil consumption can be estimated to be approximately 74.472 million barrels per day in 1997 and to be increasing at approximately 0.34 percent per year. Let $c_n$ be the daily world oil consumption (in millions of barrels per day) $n$ years after 1997. A. Find a formula for $c_n$: $c_n$=[ANS]\nB. Find $c_n-c_{n-1}$: $c_n-c_{n-1}$=[ANS]\n(Be sure that you can interpret what this difference means in practical terms.) (Be sure that you can interpret what this difference means in practical terms.)", "answer_v2": [ "74.472*(1 + 0.34/100)^n", "74.472*((1 + 0.34/100)^(n-1))*(0.34/100)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "World oil consumption can be estimated to be approximately 75.047 million barrels per day in 1999 and to be increasing at approximately 0.31 percent per year. Let $c_n$ be the daily world oil consumption (in millions of barrels per day) $n$ years after 1999. A. Find a formula for $c_n$: $c_n$=[ANS]\nB. Find $c_n-c_{n-1}$: $c_n-c_{n-1}$=[ANS]\n(Be sure that you can interpret what this difference means in practical terms.) (Be sure that you can interpret what this difference means in practical terms.)", "answer_v3": [ "75.047*(1 + 0.31/100)^n", "75.047*((1 + 0.31/100)^(n-1))*(0.31/100)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0572", "subject": "Algebra", "topic": "Finite sequences and series", "subtopic": "Notation", "level": "2", "keywords": [ "calculus", "integral", "sequences" ], "problem_v1": "Consider the sequence 9, 14, 19, 24, 29, 34... Compute the difference between successive terms and enter your answer as a list. (For example, if the sequence were 2, 5, 9, you would enter the comma separated list 3, 4 since 5-2=3 and 9-5=4). The sequence of successive differences is [ANS], which suggests that the original sequence (is/is not) [ANS] growing linearly.\nConsider the sequence 9, 14, 19, 24, 29, 34... Compute the ratio between successive terms and enter your answer as a list. (For example, if the sequence were 2, 5, 9, you would enter the comma separated list 5/2, 9/5). The sequence of successive differences is [ANS], which suggests that the original sequence (is/is not) [ANS] growing exponentially.\nFind a closed formula for the sequence 9, 14, 19, 24, 29, 34... Use $n$ as your index and start with $n=0$, that is, $b_0=9$. $b_n=$ [ANS]\nFind a closed formula for the sequence-9, 14,-19, 24,-29, 34... Use $n$ as your index and start with $n=0$, that is, $c_0=-9$. $c_n=$ [ANS]", "answer_v1": [ "(5, 5, 5, 5, 5)", "is", "(1.55556, 1.35714, 1.26316, 1.20833, 1.17241)", "is not", "5*n+9", "(-1)^(n+1)*(5*n+9)" ], "answer_type_v1": [ "OL", "MCS", "OL", "MCS", "EX", "EX" ], "options_v1": [ [], [ "is", "is not" ], [], [ "is", "is not" ], [], [] ], "problem_v2": "Consider the sequence 12, 14, 16, 18, 20, 22... Compute the difference between successive terms and enter your answer as a list. (For example, if the sequence were 2, 5, 9, you would enter the comma separated list 3, 4 since 5-2=3 and 9-5=4). The sequence of successive differences is [ANS], which suggests that the original sequence (is/is not) [ANS] growing linearly.\nConsider the sequence 12, 14, 16, 18, 20, 22... Compute the ratio between successive terms and enter your answer as a list. (For example, if the sequence were 2, 5, 9, you would enter the comma separated list 5/2, 9/5). The sequence of successive differences is [ANS], which suggests that the original sequence (is/is not) [ANS] growing exponentially.\nFind a closed formula for the sequence 12, 14, 16, 18, 20, 22... Use $n$ as your index and start with $n=0$, that is, $b_0=12$. $b_n=$ [ANS]\nFind a closed formula for the sequence-12, 14,-16, 18,-20, 22... Use $n$ as your index and start with $n=0$, that is, $c_0=-12$. $c_n=$ [ANS]", "answer_v2": [ "(2, 2, 2, 2, 2)", "is", "(1.16667, 1.14286, 1.125, 1.11111, 1.1)", "is not", "2*n+12", "(-1)^(n+1)*(2*n+12)" ], "answer_type_v2": [ "OL", "MCS", "OL", "MCS", "EX", "EX" ], "options_v2": [ [], [ "is", "is not" ], [], [ "is", "is not" ], [], [] ], "problem_v3": "Consider the sequence 9, 12, 15, 18, 21, 24... Compute the difference between successive terms and enter your answer as a list. (For example, if the sequence were 2, 5, 9, you would enter the comma separated list 3, 4 since 5-2=3 and 9-5=4). The sequence of successive differences is [ANS], which suggests that the original sequence (is/is not) [ANS] growing linearly.\nConsider the sequence 9, 12, 15, 18, 21, 24... Compute the ratio between successive terms and enter your answer as a list. (For example, if the sequence were 2, 5, 9, you would enter the comma separated list 5/2, 9/5). The sequence of successive differences is [ANS], which suggests that the original sequence (is/is not) [ANS] growing exponentially.\nFind a closed formula for the sequence 9, 12, 15, 18, 21, 24... Use $n$ as your index and start with $n=0$, that is, $b_0=9$. $b_n=$ [ANS]\nFind a closed formula for the sequence-9, 12,-15, 18,-21, 24... Use $n$ as your index and start with $n=0$, that is, $c_0=-9$. $c_n=$ [ANS]", "answer_v3": [ "(3, 3, 3, 3, 3)", "is", "(1.33333, 1.25, 1.2, 1.16667, 1.14286)", "is not", "3*n+9", "(-1)^(n+1)*(3*n+9)" ], "answer_type_v3": [ "OL", "MCS", "OL", "MCS", "EX", "EX" ], "options_v3": [ [], [ "is", "is not" ], [], [ "is", "is not" ], [], [] ] }, { "id": "Algebra_0573", "subject": "Algebra", "topic": "Finite sequences and series", "subtopic": "Notation", "level": "2", "keywords": [ "calculus", "integral", "sequences" ], "problem_v1": "The current value $f(n)$ is five times the previous value, plus four. Find a recursive definition for $f(n)$. Enter $f_{n-1}$ as $f(n-1)$. $f(n)$=[ANS]", "answer_v1": [ "5*f(n-1)+4" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "The current value $f(n)$ is two times the previous value, plus five. Find a recursive definition for $f(n)$. Enter $f_{n-1}$ as $f(n-1)$. $f(n)$=[ANS]", "answer_v2": [ "2*f(n-1)+5" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "The current value $f(n)$ is three times the previous value, plus four. Find a recursive definition for $f(n)$. Enter $f_{n-1}$ as $f(n-1)$. $f(n)$=[ANS]", "answer_v3": [ "3*f(n-1)+4" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0574", "subject": "Algebra", "topic": "Finite sequences and series", "subtopic": "Arithmetic", "level": "3", "keywords": [ "Sum", "Sigma Notation", "Closed Form" ], "problem_v1": "Express the following sum in closed form. $ \\sum\\limits_{k=1}^{n} \\ 8 \\left(\\frac{k}{n}\\right)$=[ANS]\nNote: Your answer should be in terms of $n$.", "answer_v1": [ "8*(n+1)/2" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Express the following sum in closed form. $ \\sum\\limits_{k=1}^{n} \\ 2 \\left(\\frac{k}{n}\\right)$=[ANS]\nNote: Your answer should be in terms of $n$.", "answer_v2": [ "2*(n+1)/2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Express the following sum in closed form. $ \\sum\\limits_{k=1}^{n} \\ 4 \\left(\\frac{k}{n}\\right)$=[ANS]\nNote: Your answer should be in terms of $n$.", "answer_v3": [ "4*(n+1)/2" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0575", "subject": "Algebra", "topic": "Finite sequences and series", "subtopic": "Arithmetic", "level": "3", "keywords": [ "algebra", "arithmetic sequence" ], "problem_v1": "Find $x$ such that $4x+1, \\ 4x+4,$ and $-4x-49$ are consecutive terms of an arithmetic sequence. $x=$ [ANS]", "answer_v1": [ "-7" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find $x$ such that $-8x+9, \\ 2x-3,$ and $8x-47$ are consecutive terms of an arithmetic sequence. $x=$ [ANS]", "answer_v2": [ "-8" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find $x$ such that $-4x+3, \\ 3x+1,$ and $-6x-97$ are consecutive terms of an arithmetic sequence. $x=$ [ANS]", "answer_v3": [ "-6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0576", "subject": "Algebra", "topic": "Finite sequences and series", "subtopic": "Arithmetic", "level": "2", "keywords": [ "algebra", "arithmetic sequence" ], "problem_v1": "Find the nth term of the arithmetic sequence whose initial term is $8$ and common difference is $7$. [ANS] (Your answer must be a function of $n$.)", "answer_v1": [ "8 + 7 * (n-1)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the nth term of the arithmetic sequence whose initial term is $1$ and common difference is $10$. [ANS] (Your answer must be a function of $n$.)", "answer_v2": [ "1 + 10 * (n-1)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the nth term of the arithmetic sequence whose initial term is $4$ and common difference is $7$. [ANS] (Your answer must be a function of $n$.)", "answer_v3": [ "4 + 7 * (n-1)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0577", "subject": "Algebra", "topic": "Finite sequences and series", "subtopic": "Arithmetic", "level": "3", "keywords": [ "Sequences" ], "problem_v1": "All sequences for this problem are arithmetic. Give all answers to the nearest thousandth.\nIf $a_1=50$ and $d=2$, then $a_{23}=$ [ANS].\nIf $b_{17}=-34$ and $b_{32}=14$, then $b_1=$ [ANS].\nIf $c_{16}=31$ and $c_{34}=-46$, then $S_{15}=$ [ANS].", "answer_v1": [ "94", "-85.2", "978.333" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "All sequences for this problem are arithmetic. Give all answers to the nearest thousandth.\nIf $a_1=-83$ and $d=12$, then $a_{13}=$ [ANS].\nIf $b_{13}=-37$ and $b_{49}=-64$, then $b_1=$ [ANS].\nIf $c_{13}=28$ and $c_{39}=-12$, then $S_{10}=$ [ANS].", "answer_v2": [ "61", "-28", "395.385" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "All sequences for this problem are arithmetic. Give all answers to the nearest thousandth.\nIf $a_1=-37$ and $d=3$, then $a_{15}=$ [ANS].\nIf $b_{16}=-31$ and $b_{30}=62$, then $b_1=$ [ANS].\nIf $c_{20}=-42$ and $c_{47}=-50$, then $S_{12}=$ [ANS].", "answer_v3": [ "5", "-130.643", "-456" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Algebra_0578", "subject": "Algebra", "topic": "Finite sequences and series", "subtopic": "Arithmetic", "level": "5", "keywords": [ "Sequences", "algebra", "arithmetic sequence" ], "problem_v1": "The purchase value of an office computer is 12760 dollars. Its annual depreciation is 1928 dollars. The value of the compter after 7 years is [ANS] dollars.", "answer_v1": [ "-736" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The purchase value of an office computer is 12080 dollars. Its annual depreciation is 1964 dollars. The value of the compter after 4 years is [ANS] dollars.", "answer_v2": [ "4224" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The purchase value of an office computer is 12310 dollars. Its annual depreciation is 1931 dollars. The value of the compter after 5 years is [ANS] dollars.", "answer_v3": [ "2655" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0579", "subject": "Algebra", "topic": "Finite sequences and series", "subtopic": "Arithmetic", "level": "5", "keywords": [ "Sequences" ], "problem_v1": "Determine the seating capacity of an auditorium with 35 rows of seats if there are 20 seats in the first row, 25 seats in the second row, 30 seats in the third row, 35 seats in the forth row, and so on. Total number of seats=[ANS]", "answer_v1": [ "3675" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Determine the seating capacity of an auditorium with 10 rows of seats if there are 25 seats in the first row, 27 seats in the second row, 29 seats in the third row, 31 seats in the forth row, and so on. Total number of seats=[ANS]", "answer_v2": [ "340" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Determine the seating capacity of an auditorium with 20 rows of seats if there are 20 seats in the first row, 23 seats in the second row, 26 seats in the third row, 29 seats in the forth row, and so on. Total number of seats=[ANS]", "answer_v3": [ "970" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0580", "subject": "Algebra", "topic": "Finite sequences and series", "subtopic": "Arithmetic", "level": "4", "keywords": [ "Sequences" ], "problem_v1": "Insert 5 arithmetic means between 58 and 42. First mean=[ANS]\nSecond mean=[ANS]\nThird mean=[ANS]\nFourth mean=[ANS]\nFifth mean=[ANS]\nNote: Your answers must be in decimal form, given to at least 5 places.", "answer_v1": [ "55.3333333333333", "52.6666666666667", "50", "47.3333333333333", "44.6666666666667" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Insert 5 arithmetic means between-3 and 74. First mean=[ANS]\nSecond mean=[ANS]\nThird mean=[ANS]\nFourth mean=[ANS]\nFifth mean=[ANS]\nNote: Your answers must be in decimal form, given to at least 5 places.", "answer_v2": [ "9.83333333333333", "22.6666666666667", "35.5", "48.3333333333333", "61.1666666666667" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Insert 5 arithmetic means between 18 and 45. First mean=[ANS]\nSecond mean=[ANS]\nThird mean=[ANS]\nFourth mean=[ANS]\nFifth mean=[ANS]\nNote: Your answers must be in decimal form, given to at least 5 places.", "answer_v3": [ "22.5", "27", "31.5", "36", "40.5" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Algebra_0581", "subject": "Algebra", "topic": "Finite sequences and series", "subtopic": "Arithmetic", "level": "5", "keywords": [ "logarithms", "exponentials" ], "problem_v1": "Chucky takes his first step on January 1, 2000. Every day after that, he takes 31 more steps than the day before. Tommy takes his first steps on February 1, 2000. On that day, Tommy takes 11 steps. Every day after that, Tommy takes twice as many steps as the day before. Who walks farther on Valentine's Day? [ANS]\nWho walks farther on Groundhog Day? [ANS]\nWhat is the last day in February that Chucky walks farther than Tommy? [ANS]\nNote: Your answer to parts one and two should be names. Your answer to part three should be the last day in February that Chucky takes more steps than Tommy.", "answer_v1": [ "TOMMY", "CHUCKY", "7" ], "answer_type_v1": [ "MCS", "MCS", "NV" ], "options_v1": [ [ "Tommy", "Chucky" ], [ "Tommy", "Chucky" ], [] ], "problem_v2": "Chucky takes his first step on January 1, 2000. Every day after that, he takes 4 more steps than the day before. Tommy takes his first steps on February 1, 2000. On that day, Tommy takes 14 steps. Every day after that, Tommy takes twice as many steps as the day before. Who walks farther on Valentine's Day? [ANS]\nWho walks farther on Groundhog Day? [ANS]\nWhat is the last day in February that Chucky walks farther than Tommy? [ANS]\nNote: Your answer to parts one and two should be names. Your answer to part three should be the last day in February that Chucky takes more steps than Tommy.", "answer_v2": [ "TOMMY", "CHUCKY", "4" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [ "Tommy", "Chucky" ], [ "Tommy", "Chucky" ], [] ], "problem_v3": "Chucky takes his first step on January 1, 2000. Every day after that, he takes 13 more steps than the day before. Tommy takes his first steps on February 1, 2000. On that day, Tommy takes 11 steps. Every day after that, Tommy takes twice as many steps as the day before. Who walks farther on Valentine's Day? [ANS]\nWho walks farther on Groundhog Day? [ANS]\nWhat is the last day in February that Chucky walks farther than Tommy? [ANS]\nNote: Your answer to parts one and two should be names. Your answer to part three should be the last day in February that Chucky takes more steps than Tommy.", "answer_v3": [ "TOMMY", "CHUCKY", "6" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [ "Tommy", "Chucky" ], [ "Tommy", "Chucky" ], [] ] }, { "id": "Algebra_0582", "subject": "Algebra", "topic": "Finite sequences and series", "subtopic": "Arithmetic", "level": "2", "keywords": [], "problem_v1": "Now that we have looked at the Gauss Trick, we would like to see why it works. Mathematicians do this sort of thing to see whether a trick like this will work in more complicated situations. So we will ask a few true-false questions to see if you can figure out why it works. [ANS] 1. The Gauss Trick will only work when the differences between consecutive numbers being summed is the same [ANS] 2. To find the sum using the Gauss trick, we always multiply the column sums by 1 more than the larger number and then divide by 2 [ANS] 3. To find the sum using the Gauss trick, we always multiply the column sums by the difference between the largest and smallest number and then divide by 2 [ANS] 4. The Gauss Trick will only work when the numbers being added are consecutive (that is, we do not skip any numbers between the first and the last.) [ANS] 5. To find the sum using the Gauss trick, we always multiply the column sums by the number of numbers being summed and then divide by 2 [ANS] 6. The Gauss Trick will only work when the column sums (that is, the sums of the pairs consisting of a number in the first line and the number below it in the second line) are all the same number", "answer_v1": [ "T", "F", "F", "F", "T", "T" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v1": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v2": "Now that we have looked at the Gauss Trick, we would like to see why it works. Mathematicians do this sort of thing to see whether a trick like this will work in more complicated situations. So we will ask a few true-false questions to see if you can figure out why it works. [ANS] 1. The Gauss Trick will only work when the numbers being added are consecutive (that is, we do not skip any numbers between the first and the last.) [ANS] 2. To find the sum using the Gauss trick, we always multiply the column sums by 1 more than the larger number and then divide by 2 [ANS] 3. To find the sum using the Gauss trick, we always multiply the column sums by the difference between the largest and smallest number and then divide by 2 [ANS] 4. The Gauss Trick will only work when the column sums (that is, the sums of the pairs consisting of a number in the first line and the number below it in the second line) are all the same number [ANS] 5. To find the sum using the Gauss trick, we always multiply the column sums by the number of numbers being summed and then divide by 2 [ANS] 6. The Gauss Trick will only work when the differences between consecutive numbers being summed is the same", "answer_v2": [ "F", "F", "F", "T", "T", "T" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v2": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v3": "Now that we have looked at the Gauss Trick, we would like to see why it works. Mathematicians do this sort of thing to see whether a trick like this will work in more complicated situations. So we will ask a few true-false questions to see if you can figure out why it works. [ANS] 1. The Gauss Trick will only work when the differences between consecutive numbers being summed is the same [ANS] 2. The Gauss Trick will only work when the numbers being added are consecutive (that is, we do not skip any numbers between the first and the last.) [ANS] 3. The Gauss Trick will only work when the column sums (that is, the sums of the pairs consisting of a number in the first line and the number below it in the second line) are all the same number [ANS] 4. To find the sum using the Gauss trick, we always multiply the column sums by the difference between the largest and smallest number and then divide by 2 [ANS] 5. To find the sum using the Gauss trick, we always multiply the column sums by 1 more than the larger number and then divide by 2 [ANS] 6. To find the sum using the Gauss trick, we always multiply the column sums by the number of numbers being summed and then divide by 2", "answer_v3": [ "T", "F", "T", "F", "F", "T" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v3": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ] }, { "id": "Algebra_0583", "subject": "Algebra", "topic": "Finite sequences and series", "subtopic": "Arithmetic", "level": "2", "keywords": [ "algebra", "sequence" ], "problem_v1": "Find the following partial sum,\n$ \\sum_{n=0}^{40} (3-2 n)=$ [ANS]", "answer_v1": [ "(40+1) *(2*3+40*-2)/2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the following partial sum,\n$ \\sum_{n=0}^{20} (5-5 n)=$ [ANS]", "answer_v2": [ "(20+1) *(2*5+20*-5)/2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the following partial sum,\n$ \\sum_{n=0}^{30} (4-4 n)=$ [ANS]", "answer_v3": [ "(30+1) *(2*4+30*-4)/2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0584", "subject": "Algebra", "topic": "Finite sequences and series", "subtopic": "Arithmetic", "level": "4", "keywords": [ "algebra", "sequences", "arithmetic sequence", "series" ], "problem_v1": "Find the sum of the first $325$ positive even whole numbers. Sum: [ANS]", "answer_v1": [ "325*(1+325)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the sum of the first $125$ positive even whole numbers. Sum: [ANS]", "answer_v2": [ "125*(1+125)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the sum of the first $175$ positive even whole numbers. Sum: [ANS]", "answer_v3": [ "175*(1+175)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0585", "subject": "Algebra", "topic": "Finite sequences and series", "subtopic": "Arithmetic", "level": "5", "keywords": [ "algebra", "sequences", "arithmetic", "series", "word problems" ], "problem_v1": "Sam is saving quarters. She saves $1$ quarter the first day, $2$ quarters the second day, $3$ quarters the third day, and so on for $30$ days. How much money will she have saved in $30$ days? Answer (in dollars): $ [ANS]", "answer_v1": [ "116.25" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Sam is saving quarters. She saves $1$ quarter the first day, $2$ quarters the second day, $3$ quarters the third day, and so on for $20$ days. How much money will she have saved in $20$ days? Answer (in dollars): $ [ANS]", "answer_v2": [ "52.50" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Sam is saving quarters. She saves $1$ quarter the first day, $2$ quarters the second day, $3$ quarters the third day, and so on for $28$ days. How much money will she have saved in $28$ days? Answer (in dollars): $ [ANS]", "answer_v3": [ "101.50" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0586", "subject": "Algebra", "topic": "Finite sequences and series", "subtopic": "Arithmetic", "level": "5", "keywords": [ "algebra", "sequences", "arithmetic", "series", "word problems" ], "problem_v1": "An object falling from rest in a vacuum falls approximately $16$ feet the first second, $48$ feet the second, $80$ feet the third second, $112$ feet the fourth second, and so on. How far will it fall in $12$ seconds? Answer (in feet): [ANS]", "answer_v1": [ "2304" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "An object falling from rest in a vacuum falls approximately $16$ feet the first second, $48$ feet the second, $80$ feet the third second, $112$ feet the fourth second, and so on. How far will it fall in $7$ seconds? Answer (in feet): [ANS]", "answer_v2": [ "784" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "An object falling from rest in a vacuum falls approximately $16$ feet the first second, $48$ feet the second, $80$ feet the third second, $112$ feet the fourth second, and so on. How far will it fall in $9$ seconds? Answer (in feet): [ANS]", "answer_v3": [ "1296" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0587", "subject": "Algebra", "topic": "Finite sequences and series", "subtopic": "Arithmetic", "level": "5", "keywords": [ "algebra", "sequences", "arithmetic", "series", "word problems" ], "problem_v1": "A pile of logs has $45$ logs in the bottom layer, $44$ logs in the next layer, $43$ logs in the next layer, and so on, until the top layer has $1$ log. How many logs are in the pile? Answer (in logs): [ANS]", "answer_v1": [ "1035" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A pile of logs has $25$ logs in the bottom layer, $24$ logs in the next layer, $23$ logs in the next layer, and so on, until the top layer has $1$ log. How many logs are in the pile? Answer (in logs): [ANS]", "answer_v2": [ "325" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A pile of logs has $30$ logs in the bottom layer, $29$ logs in the next layer, $28$ logs in the next layer, and so on, until the top layer has $1$ log. How many logs are in the pile? Answer (in logs): [ANS]", "answer_v3": [ "465" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0588", "subject": "Algebra", "topic": "Finite sequences and series", "subtopic": "Arithmetic", "level": "2", "keywords": [ "prealgebra", "common core" ], "problem_v1": "Find the next three numbers in the pattern:\n$5,-10,20,-40,$ [ANS]\n(Separate your answers by commas.)", "answer_v1": [ "(80, -160, 320)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find the next three numbers in the pattern:\n$-7,14,-28,56,$ [ANS]\n(Separate your answers by commas.)", "answer_v2": [ "(-112, 224, -448)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find the next three numbers in the pattern:\n$-5,10,-20,40,$ [ANS]\n(Separate your answers by commas.)", "answer_v3": [ "(-80, 160, -320)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Algebra_0589", "subject": "Algebra", "topic": "Finite sequences and series", "subtopic": "Binomial theorem", "level": "2", "keywords": [ "algebra", "polynomials", "binomial" ], "problem_v1": "Write the first four terms of the binomial expansion of $(a-4 b)^{14}.$ Answer: [ANS]", "answer_v1": [ "a^14-56*a^13*b+1456*a^12*b^2-23296*a^11*b^3" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Write the first four terms of the binomial expansion of $(a-2 b)^{16}.$ Answer: [ANS]", "answer_v2": [ "a^16-32*a^15*b+480*a^14*b^2-4480*a^13*b^3" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Write the first four terms of the binomial expansion of $(a-2 b)^{15}.$ Answer: [ANS]", "answer_v3": [ "a^15-30*a^14*b+420*a^13*b^2-3640*a^12*b^3" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0590", "subject": "Algebra", "topic": "Finite sequences and series", "subtopic": "Summation formulas", "level": "3", "keywords": [ "algebra", "sequence" ], "problem_v1": "Wirte the sum using sigma notation: $\\frac{1}{1\\cdot 2}+\\frac{1}{2\\cdot 3}+\\frac{1}{3\\cdot 4}+\\frac{1}{201\\cdot 202}=\\sum_{n=1}^{A} B$, where $A=$ [ANS], $B=$ [ANS].", "answer_v1": [ "201", "1/(n*(n+1))" ], "answer_type_v1": [ "NV", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Wirte the sum using sigma notation: $\\frac{1}{1\\cdot 2}+\\frac{1}{2\\cdot 3}+\\frac{1}{3\\cdot 4}+\\frac{1}{93\\cdot 94}=\\sum_{n=1}^{A} B$, where $A=$ [ANS], $B=$ [ANS].", "answer_v2": [ "93", "1/(n*(n+1))" ], "answer_type_v2": [ "NV", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Wirte the sum using sigma notation: $\\frac{1}{1\\cdot 2}+\\frac{1}{2\\cdot 3}+\\frac{1}{3\\cdot 4}+\\frac{1}{130\\cdot 131}=\\sum_{n=1}^{A} B$, where $A=$ [ANS], $B=$ [ANS].", "answer_v3": [ "130", "1/(n*(n+1))" ], "answer_type_v3": [ "NV", "EX" ], "options_v3": [ [], [] ] }, { "id": "Algebra_0591", "subject": "Algebra", "topic": "Conic sections", "subtopic": "Parabolas", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Find the equation of the parabola with the given properties Vertex $(0,0)$, focus $(8,0)$. [ANS] $=x$", "answer_v1": [ "y^2/32" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the equation of the parabola with the given properties Vertex $(0,0)$, focus $(2,0)$. [ANS] $=x$", "answer_v2": [ "y^2/8" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the equation of the parabola with the given properties Vertex $(0,0)$, focus $(4,0)$. [ANS] $=x$", "answer_v3": [ "y^2/16" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0592", "subject": "Algebra", "topic": "Conic sections", "subtopic": "Ellipses", "level": "3", "keywords": [ "Conic", "Ellipse", "Center", "Vertex", "Focus", "ellipse", "conics" ], "problem_v1": "Find the center, vertices, and foci of each ellipse.\n(a) $ \\frac{x^2}{81}+\\frac{y^2}{16}=1$ Center: ([ANS], [ANS]) Right vertex: ([ANS], [ANS]) Left vertex: ([ANS], [ANS]) Top vertex: ([ANS], [ANS]) Bottom vertex: ([ANS], [ANS]) Right focus: ([ANS], [ANS]) Left focus: ([ANS], [ANS])\n(b) $ \\frac{(x+7)^2}{16}+\\frac{(y-7)^2}{81}=1$ Center: ([ANS], [ANS]) Right vertex: ([ANS], [ANS]) Left vertex: ([ANS], [ANS]) Top vertex: ([ANS], [ANS]) Bottom vertex: ([ANS], [ANS]) Top focus: ([ANS], [ANS]) Bottom focus: ([ANS], [ANS])\n(c) $9x^2+16 y^2-108x-128 y+436=0$ Center: ([ANS], [ANS]) Right vertex: ([ANS], [ANS]) Left vertex: ([ANS], [ANS]) Top vertex: ([ANS], [ANS]) Bottom vertex: ([ANS], [ANS]) Right focus: ([ANS], [ANS]) Left focus: ([ANS], [ANS])", "answer_v1": [ "0", "0", "9", "0", "-9", "0", "0", "4", "0", "-4", "8.06225774829855", "0", "-8.06225774829855", "0", "-7", "7", "-3", "7", "-11", "7", "-7", "16", "-7", "-2", "-7", "15.0622577482985", "-7", "-1.06225774829855", "6", "4", "10", "4", "2", "4", "6", "7", "6", "1", "8.64575131106459", "4", "3.35424868893541", "4" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Find the center, vertices, and foci of each ellipse.\n(a) $ \\frac{x^2}{36}+\\frac{y^2}{25}=1$ Center: ([ANS], [ANS]) Right vertex: ([ANS], [ANS]) Left vertex: ([ANS], [ANS]) Top vertex: ([ANS], [ANS]) Bottom vertex: ([ANS], [ANS]) Right focus: ([ANS], [ANS]) Left focus: ([ANS], [ANS])\n(b) $ \\frac{(x+19)^2}{4}+\\frac{(y-7)^2}{49}=1$ Center: ([ANS], [ANS]) Right vertex: ([ANS], [ANS]) Left vertex: ([ANS], [ANS]) Top vertex: ([ANS], [ANS]) Bottom vertex: ([ANS], [ANS]) Top focus: ([ANS], [ANS]) Bottom focus: ([ANS], [ANS])\n(c) $4x^2+9 y^2-32x-108 y+352=0$ Center: ([ANS], [ANS]) Right vertex: ([ANS], [ANS]) Left vertex: ([ANS], [ANS]) Top vertex: ([ANS], [ANS]) Bottom vertex: ([ANS], [ANS]) Right focus: ([ANS], [ANS]) Left focus: ([ANS], [ANS])", "answer_v2": [ "0", "0", "6", "0", "-6", "0", "0", "5", "0", "-5", "3.3166247903554", "0", "-3.3166247903554", "0", "-19", "7", "-17", "7", "-21", "7", "-19", "14", "-19", "0", "-19", "13.7082039324994", "-19", "0.291796067500631", "4", "6", "7", "6", "1", "6", "4", "8", "4", "4", "6.23606797749979", "6", "1.76393202250021", "6" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Find the center, vertices, and foci of each ellipse.\n(a) $ \\frac{x^2}{49}+\\frac{y^2}{16}=1$ Center: ([ANS], [ANS]) Right vertex: ([ANS], [ANS]) Left vertex: ([ANS], [ANS]) Top vertex: ([ANS], [ANS]) Bottom vertex: ([ANS], [ANS]) Right focus: ([ANS], [ANS]) Left focus: ([ANS], [ANS])\n(b) $ \\frac{(x+5)^2}{9}+\\frac{(y-7)^2}{64}=1$ Center: ([ANS], [ANS]) Right vertex: ([ANS], [ANS]) Left vertex: ([ANS], [ANS]) Top vertex: ([ANS], [ANS]) Bottom vertex: ([ANS], [ANS]) Top focus: ([ANS], [ANS]) Bottom focus: ([ANS], [ANS])\n(c) $9x^2+16 y^2-180x-288 y+2052=0$ Center: ([ANS], [ANS]) Right vertex: ([ANS], [ANS]) Left vertex: ([ANS], [ANS]) Top vertex: ([ANS], [ANS]) Bottom vertex: ([ANS], [ANS]) Right focus: ([ANS], [ANS]) Left focus: ([ANS], [ANS])", "answer_v3": [ "0", "0", "7", "0", "-7", "0", "0", "4", "0", "-4", "5.74456264653803", "0", "-5.74456264653803", "0", "-5", "7", "-2", "7", "-8", "7", "-5", "15", "-5", "-1", "-5", "14.4161984870957", "-5", "-0.416198487095663", "10", "9", "14", "9", "6", "9", "10", "12", "10", "6", "12.6457513110646", "9", "7.35424868893541", "9" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Algebra_0593", "subject": "Algebra", "topic": "Conic sections", "subtopic": "Ellipses", "level": "3", "keywords": [ "calculus", "conic section", "ellipse", "parabola", "hyperbola" ], "problem_v1": "Consider the conic section given by the equation 204849x^2-68040xy+137376 y^2-1661634x-1710072 y+8884161=0 Then an appropriate rotation of coordinate axes to eliminate the $xy$ term is given by the equations $x=$ [ANS] $X+$ [ANS] $Y$ $y=$ [ANS] $X+$ [ANS] $Y$ After applying this rotation, we obtain the following equation in $X$ and $Y$ [ANS]\n(Do NOT simplify the equation you get by multiplying or dividing by any factor.) Which conic section is it? (Acceptable answers are: ellipse, hyperbola and parabola.) Answer: [ANS]\nThe first focus of this conic has $xy$ coordinates ([ANS], [ANS]). (Order the foci lexicographically according to the order of their $x$ and $y$ coordinates, ie. $(-1,5)$ precedes $(3,-2)$ and $(2,1)$ precedes $(2,4)$.) The second focus of this conic has $xy$ coordinates ([ANS], [ANS]). (If the conic is a parabola, just repeat the coordinates of the first focus.) The equation of the directrix is [ANS] $=0$. (If the conic is a parabola, write the directrix in the form $X-p=0$ or $Y-p=0$, then apply an appropriate rotation of axes to get an equation in $x$ and $y$. If the conic is an ellipse or hyperbola, write the equation $1=0$, an impossible equation.) The axis of the conic has equation [ANS] $=0$. (The axis of a conic is the line joining the foci and the vertices. For an ellipse this is also known as the major axis. Write the equation in the form $X-c=0$ or $Y-c=0$, then apply an appropriate rotation of axes to get an equation in $x$ and $y$.) The asymptote of smaller slope has equation $y=$ [ANS]. (If the conic is not a hyperbola put $y+1$ on the right hand side of the equation, giving an impossible equation.) The asymptote of larger slope has equation $y=$ [ANS]. (If the conic is not a hyperbola put $y+1$ on the right hand side of the equation, giving an impossible equation.)", "answer_v1": [ "0.923076923076923", "0.384615384615385", "-0.384615384615385", "0.923076923076923", "117^2*(16*x^2+9*y^2+-64*x+-162*y+793- 144)", "ellipse", "4.29009564959054", "5.0962295590173", "6.32528896579407", "9.98069351790578", "1", "(108/117)*x+(-45/117)*y - 2", "y+1", "y+1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "EX", "MCS", "NV", "NV", "NV", "NV", "NV", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [ "ellipse", "hyperbola", "parabola" ], [], [], [], [], [], [], [], [] ], "problem_v2": "Consider the conic section given by the equation 400192x^2-266112xy+177808 y^2-6212480x+4180640 y+25085200=0 Then an appropriate rotation of coordinate axes to eliminate the $xy$ term is given by the equations $x=$ [ANS] $X+$ [ANS] $Y$ $y=$ [ANS] $X+$ [ANS] $Y$ After applying this rotation, we obtain the following equation in $X$ and $Y$ [ANS]\n(Do NOT simplify the equation you get by multiplying or dividing by any factor.) Which conic section is it? (Acceptable answers are: ellipse, hyperbola and parabola.) Answer: [ANS]\nThe first focus of this conic has $xy$ coordinates ([ANS], [ANS]). (Order the foci lexicographically according to the order of their $x$ and $y$ coordinates, ie. $(-1,5)$ precedes $(3,-2)$ and $(2,1)$ precedes $(2,4)$.) The second focus of this conic has $xy$ coordinates ([ANS], [ANS]). (If the conic is a parabola, just repeat the coordinates of the first focus.) The equation of the directrix is [ANS] $=0$. (If the conic is a parabola, write the directrix in the form $X-p=0$ or $Y-p=0$, then apply an appropriate rotation of axes to get an equation in $x$ and $y$. If the conic is an ellipse or hyperbola, write the equation $1=0$, an impossible equation.) The axis of the conic has equation [ANS] $=0$. (The axis of a conic is the line joining the foci and the vertices. For an ellipse this is also known as the major axis. Write the equation in the form $X-c=0$ or $Y-c=0$, then apply an appropriate rotation of axes to get an equation in $x$ and $y$.) The asymptote of smaller slope has equation $y=$ [ANS]. (If the conic is not a hyperbola put $y+1$ on the right hand side of the equation, giving an impossible equation.) The asymptote of larger slope has equation $y=$ [ANS]. (If the conic is not a hyperbola put $y+1$ on the right hand side of the equation, giving an impossible equation.)", "answer_v2": [ "0.905882352941176", "0.423529411764706", "-0.423529411764706", "0.905882352941176", "85^2*(64*x^2+16*Y^2+-1024*x+160*y+4496- 1024)", "ellipse", "2.19511392600096", "-14.1937841027202", "8.0637096034108", "-1.64151001492689", "1", "(77/85)*x+(-36/85)*y - 8", "y+1", "y+1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "EX", "MCS", "NV", "NV", "NV", "NV", "NV", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [ "ellipse", "hyperbola", "parabola" ], [], [], [], [], [], [], [], [] ], "problem_v3": "Consider the conic section given by the equation 130320x^2+73920xy+89380 y^2-954720x-1974960 y+8517600=0 Then an appropriate rotation of coordinate axes to eliminate the $xy$ term is given by the equations $x=$ [ANS] $X+$ [ANS] $Y$ $y=$ [ANS] $X+$ [ANS] $Y$ After applying this rotation, we obtain the following equation in $X$ and $Y$ [ANS]\n(Do NOT simplify the equation you get by multiplying or dividing by any factor.) Which conic section is it? (Acceptable answers are: ellipse, hyperbola and parabola.) Answer: [ANS]\nThe first focus of this conic has $xy$ coordinates ([ANS], [ANS]). (Order the foci lexicographically according to the order of their $x$ and $y$ coordinates, ie. $(-1,5)$ precedes $(3,-2)$ and $(2,1)$ precedes $(2,4)$.) The second focus of this conic has $xy$ coordinates ([ANS], [ANS]). (If the conic is a parabola, just repeat the coordinates of the first focus.) The equation of the directrix is [ANS] $=0$. (If the conic is a parabola, write the directrix in the form $X-p=0$ or $Y-p=0$, then apply an appropriate rotation of axes to get an equation in $x$ and $y$. If the conic is an ellipse or hyperbola, write the equation $1=0$, an impossible equation.) The axis of the conic has equation [ANS] $=0$. (The axis of a conic is the line joining the foci and the vertices. For an ellipse this is also known as the major axis. Write the equation in the form $X-c=0$ or $Y-c=0$, then apply an appropriate rotation of axes to get an equation in $x$ and $y$.) The asymptote of smaller slope has equation $y=$ [ANS]. (If the conic is not a hyperbola put $y+1$ on the right hand side of the equation, giving an impossible equation.) The asymptote of larger slope has equation $y=$ [ANS]. (If the conic is not a hyperbola put $y+1$ on the right hand side of the equation, giving an impossible equation.)", "answer_v3": [ "0.861538461538462", "-0.507692307692308", "0.507692307692308", "0.861538461538462", "65^2*(36*X^2+16*Y^2+-432*X+-288*Y+2592- 576)", "ellipse", "-1.67046902330748", "14.6529171304612", "2.87046902330748", "6.94708286953882", "1", "(56/65)*x+(33/65)*y - 6", "y+1", "y+1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "EX", "MCS", "NV", "NV", "NV", "NV", "NV", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [ "ellipse", "hyperbola", "parabola" ], [], [], [], [], [], [], [], [] ] }, { "id": "Algebra_0594", "subject": "Algebra", "topic": "Conic sections", "subtopic": "Ellipses", "level": "5", "keywords": [ "calculus" ], "problem_v1": "A bridge underpass in the shape of an elliptical arch, that is, half of an ellipse, is $50$ feet wide and $14$ feet high. An eight foot wide rectangular truck is to drive (safely) underneath. How high can it be?\n$h=$ [ANS]", "answer_v1": [ "13.8196382007634" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A bridge underpass in the shape of an elliptical arch, that is, half of an ellipse, is $20$ feet wide and $15$ feet high. An eight foot wide rectangular truck is to drive (safely) underneath. How high can it be?\n$h=$ [ANS]", "answer_v2": [ "13.7477270848675" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A bridge underpass in the shape of an elliptical arch, that is, half of an ellipse, is $30$ feet wide and $14$ feet high. An eight foot wide rectangular truck is to drive (safely) underneath. How high can it be?\n$h=$ [ANS]", "answer_v3": [ "13.4930434751476" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Algebra_0595", "subject": "Algebra", "topic": "Conic sections", "subtopic": "Ellipses", "level": "3", "keywords": [ "calculus" ], "problem_v1": "The equation of an ellipse with center $(5,3)$ that passes through the points $(9,3)$ and $(5,5)$ has the form $f(x,y)=1$. Find $f(x,0)$. [ANS]", "answer_v1": [ " (x-5)^{2} /16 + 9/4" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "The equation of an ellipse with center $(2,3)$ that passes through the points $(6,3)$ and $(2,5)$ has the form $f(x,y)=1$. Find $f(x,0)$. [ANS]", "answer_v2": [ " (x-2)^{2} /16 + 9/4" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "The equation of an ellipse with center $(3,3)$ that passes through the points $(7,3)$ and $(3,5)$ has the form $f(x,y)=1$. Find $f(x,0)$. [ANS]", "answer_v3": [ " (x-3)^{2} /16 + 9/4" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Algebra_0596", "subject": "Algebra", "topic": "Conic sections", "subtopic": "Polar or parametric form", "level": "3", "keywords": [ "Conics", "Polar coordinates" ], "problem_v1": "Find a polar equation for the ellipse that has its focus at the pole and satisfies the stated conditions. Enter t for the variable $\\theta$.\n(a) Directrix to the right of the pole; $b=$ 5 and $e=\\frac{3}{5}$. Equation of the ellipse is $r=$ [ANS]\n(b) Directrix below the pole; $c=$ 6 and $e=\\frac{1}{4}$. Equation of the ellipse is $r=$ [ANS]", "answer_v1": [ "20/[5+3*cos(t)]", "90/[4-sin(t)]" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Find a polar equation for the ellipse that has its focus at the pole and satisfies the stated conditions. Enter t for the variable $\\theta$.\n(a) Directrix above the pole; $b=$ 7 and $e=\\frac{3}{5}$. Equation of the ellipse is $r=$ [ANS]\n(b) Directrix to the left of the pole; $c=$ 3 and $e=\\frac{1}{3}$. Equation of the ellipse is $r=$ [ANS]", "answer_v2": [ "28/[5+3*sin(t)]", "24/[3-cos(t)]" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Find a polar equation for the ellipse that has its focus at the pole and satisfies the stated conditions. Enter t for the variable $\\theta$.\n(a) Directrix below the pole; $b=$ 5 and $e=\\frac{3}{5}$. Equation of the ellipse is $r=$ [ANS]\n(b) Directrix to the right of the pole; $c=$ 3 and $e=\\frac{1}{4}$. Equation of the ellipse is $r=$ [ANS]", "answer_v3": [ "20/[5-3*sin(t)]", "45/[4+cos(t)]" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] } ]