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Math 1 EOC Practice Test Answer Key: Part 1: 1. B 2. C 3. B 4. D 5. D 6. 120 7. 0.75 8. 5 9. 0 10. 10 11. -5 12. 16 13. 9 14. 4 15. 1 Part 2: 1. B 2. C 3. D 4. C 5. B 6. D 7. C 8. B 9. D 10. B 11. D 12. A 13. B 14. D 15. B 16. D 17. C 18. B 19. A 20. C 21. C 22. D 23. C 24. A 25. A 26. A 27. C 28. C 29. C 30. B 31. A 32. C 33. C 34. B 35. A Part 1 Calculators are not allowed on Part 1. 2 1. Jessie’s bus ride to school is 5 minutes more than 3 the time of Robert’s bus ride. Which graph shows the possible times of Jessie’s and Robert’s bus rides? Because the answer choices are graphs of lines, we know that we are looking for a linear equation. Remember: linear equations take the form π¦ = ππ₯ + π, where m is the slope and b is the y-intercept. Also, notice that in each graph Robert’s bus ride is graphed along the x-axis and Jessie’s bus ride is graphed along the y-axis. This means we’ll need to let x represent Robert’s bus ride and y represent Jessie’s. 2 As an equation, we can write “Jessie’s bus ride to school is 5 minutes more than 3 the π time of Robert’s bus ride” as π = π π + π. The correct answer choice will be a graph with 2 a slope of 3 and a y-intercept of 5. B is the answer. a) b) 2. What scenario could be modeled by the graph below? The graph is our clue that we are dealing with a linear inequality here—linear because we have a straight line and an inequality (as opposed to an equation) because we have a shaded region below the line. To solve, we need to find which of the answer choices matches the graph. Notice that the slope is negative and the y-intercept is 5. Checking “rise over run,” we can also find that the slope is −2. We need to make use of this information to find the right answer below. a) The number of pounds of apples, y, minus two times the number of pounds of oranges, x, is at most 5. This answer gives the inequality: π¦ − 2π₯ ≤ 5. After we add −2π₯ to both sides, we get π¦ ≤ 2π₯ + 5, which has the wrong slope. b) The number of pounds of apples, y, minus half the number of pounds of oranges, x, is at most 5. c) The number of pounds of apples, y, plus two times the number of pounds of oranges, x, is at most 5. This answer gives the inequality: π¦ + 2π₯ ≤ 5. After we subtract 2π₯ from both sides, we get π¦ ≤ −2π₯ + 5, which matches the graph above. d) The number of pounds of apples, y, plus half the number of pounds of oranges, x, is at most 5. 3. Which expression is equivalent to π‘ 2 − 36? A “difference of squares” problem: both terms are perfect squares and the second is being subtracted from the first. We need to find factors that, when FOILed, will cause the middle term to cancel out. a) (π‘ − 6)(π‘ − 6) When this is FOILed, we get π‘ 2 − 12π‘ + 36. b) (π‘ + 6)(π‘ − 6) When this is FOILed, we get π‘ 2 + ππ − ππ − 36, which simplifies to π‘ 2 − 36. c) (π‘ − 12)(π‘ − 3) d) (π‘ − 12)(π‘ + 3) 4. Which graph represents the function π¦ = 3π₯ 2 + 12π₯ − 6? Because this function has a degree of 2 (the power of the first term is 2), we know that it is a quadratic function. Quadratic functions have U-shaped graphs known as parabolas. A few important rules to keep in mind for graphs of quadratic functions: 1) Positive functions, like the one in this problem, have u-shaped graphs (think: positive ο smiley face); negative functions have graphs that point down in the shape of a lower-case ‘n’ (think: negative ο frowning face). This rule alone allows us to eliminate F and I as possible answer choices. 2) We can find the axis of symmetry (the vertical line that divides the parabola in 2) for a quadratic graph using π = −π ππ . (Note: quadratics take the form ax 2 + bx + c, so a is the leading coefficient and b is the coefficient of the middle term.) Here, the axis of symmetry is −12 6 , or −2. This means we can also eliminate H as an answer choice. The answer is G. 3) It is also helpful to know that the y-intercept is determined by c, the third term in the function. So in this problem the y-intercept is -6. 5. The floor of a rectangular cage has a length 4 feet greater than its width, w. James will increase both dimensions of the floor by two feet. Which equation represents the new area, N, of the floor of the cage? π€ π₯+4 If both dimensions are increased by 2 feet, the dimensions become π€ + 2 and π€ + 6. To find the new area we can use π΄πππ = πππππ‘β β π€πππ‘β. We simply need to multiply (π€ + 2) and (π€ + 6). Using FOIL (first, outer, inner, last) to multiply the 2 binomials, we get π€ 2 + 6π€ + 2π€ + 12, which simplifies to π€ 2 + 8π€ + 12. a) π = π€ 2 + 4π€ b) π = π€ 2 + 6π€ c) π = π€ 2 + 6π€ + 8 d) π = π€ 2 + 8π€ + 12 6. Two boys, Shawn and Curtis, went for a walk. Shawn began walking 20 seconds earlier than Curtis. ο· ο· Shawn walked at a speed of 5 feet per second. Curtis walked at a speed of 6 feet per second. For how many seconds had Shawn been walking at the moment when the two boys had walked exactly the same distance? Another linear equations problem. First, we need to write an equation for both Shawn and Curtis: Shawn: π¦ = 5π₯ + 20 Curtis: π¦ = 6π₯ Now, because we’re trying to find when the boys had walked the same distance, we need to determine where the two graphs intersect. One option would be to graph both lines to find the point of intersection. However, we can save time by setting the equations equal to one another and solving for x: 5π₯ + 20 = 6π₯ π₯ = 20 20 is not the answer, however; 20 is x. To solve, we have to substitute 20 for x in either equation. Doing so, we find that π¦ = 120. 7. The math club sells candy bars and drinks during the football games. ο· ο· 60 candy bars and 110 drinks will sell for $265. 120 candy bars and 90 drinks will sell for $270. How much does each candy bar sell for? This is a system of equations problem. We need to write two equations and then solve, either by elimination or substitution, for the cost of candy bars and drinks. Based on the information given, we can write the following equations: 60π + 110π = 265 120π + 90π = 270 By multiplying the entire first equation by 2, we can solve by elimination: 2(60π + 110π = 265) = ππππ + ππππ = πππ The purpose of doing this is to get identical terms in each equation which can be eliminated by subtracting: 120π + 220π = 530 −120π + 90π = 270 130π = 260 ο **Remember to change the sign of each term when subtracting polynomials. π=2 Now that we know drinks cost $2, we can substitute 2 for d in either equation and solve for the cost of candy bars, c: 120π + 220(2) = 530 = 120π + 440 = 530 = 120π = 90 90 = π = 120 = π = $0.75 8. What is the smallest of 3 consecutive positive integers if the product of the smallest two integers is 5 less than 5 times the largest integer? Examples of 3 consecutive positive integers would be 1, 2, 3 or 17, 18, 19. This means that if we call the first integer π₯, the next two would have to equal π₯ + 1 and π₯ + 2. Using these expressions we can write the equation as described in the problem: π₯(π₯ + 1) = 5(π₯ + 2) − 5 Now, we have to solve for x: π₯(π₯ + 1) = 5(π₯ + 2) − 5 π₯ 2 + π₯ = 5π₯ + 10 − 5 π₯ 2 + π₯ = 5π₯ + 5 π₯ 2 − 4π₯ − 5 = 0 Factoring the trinomial gives us (π₯ − 5)(π₯ + 1) = 0. Now we set each factor equal to 0 and solve for x: π₯−5=0 π₯+1=0 π₯=5 π₯ = −1 Because the problem specified “three consecutive positive integers,” π₯ = −1 doesn’t fit the description. The three terms are 5, 6, 7. 9. The function π(π‘) = −5π‘ 2 + 20π‘ + 60 models the height of an object t seconds after it is launched. How many seconds does it take the object to hit the ground? As we saw in problem 4 above, negative quadratic functions like this one have downward-facing, n-shaped graphs. Such a graph models an object as it is launched into the air and returns to the ground. Keeping in mind that t represents the number of seconds, we know that f(t) represents the height of the object. This means that t will have to be a number that makes f(t) equal zero. First, factor the quadratic: −5π‘ 2 + 20π‘ + 60 −5(π‘ 2 − 4π‘ − 12) Pull out the GCF, -5. −5(π‘ − 6)(π‘ + 2) Factor the remaining trinomial. π‘−6=0 π‘=6 π‘+2=0 Set each factor equal to 0. π‘ = −2 Solve for t. Since -2 doesn’t make sense for an amount of time, 6 is the answer. 10. Two times Antonio’s age plus three times Sarah’s age equals 34. Sarah’s age is also five times Antonio’s age. How old is Sarah? Another system of equations problem. Begin by writing two equations: 2π + 3π = 34 π = 5π Because we already have one of our variables, s, isolated, this system is easily solved by substitution. All we need to do is substitute 5a, which we know equals s, into the first equation for s. So, 2π + 3π = 34 becomes 2π + 3(5π) = 34 Simplify: 2π + 3(5π) = 34 2π + 15π = 34 17π = 34 π=2 The problem asks us to solve for Sarah’s age, so we finish up by substituting 2 for a into either of the original equations and solving for s: π = 5π π = 5(2) π = 10 11. The function π(π₯) = 2(2)π₯ was replaced with π(π₯) + π, resulting in the function graphed below. What is the value of k? Adding or subtracting a number to an exponential function results in a vertical shift to its graph. When a number is added, the graph shifts upward that exact amount. Likewise, when a number is subtracted, the graph shifts downward that exact amount. Here, since the graph has been shifted down 5 places from the x-axis (notice that the graph begins along the line π¦ = −5), k has to equal −5. 12. Suppose that the function π(π₯) = 2π₯ + 12 represents the cost to rent x movies a month from an internet club. Makayla now has $10. How many more dollars does Makayla need to rent 7 movies next month? Remember: functions involve inputs and outputs; we plug in one number and get back another. By plugging 7 in for x, this particular function will tell us how much it costs to rent 7 movies: π(7) = 2(7) + 12 π(7) = 14 + 12 π(7) = 26 It costs $26 to rent 7 movies. Since Makayla has $10, she needs $16 more to rent 7 movies. 13. The larger leg of a right triangle is 3cm longer than its smaller leg. The hypotenuse is 6cm longer than the smaller leg. How many centimeters long is the smaller leg? Whenever we know the lengths of any two sides of a right triangle, we can solve for the third side using the Pythagorean Theorem: ππ + ππ = ππ . Just remember that in the Pythagorean Theorem, c is always the hypotenuse, which is the longest side of a triangle. In this problem, we don’t have numbers to plug into the Pythagorean Theorem, but we can express the sides of the triangle algebraically based on the information in the problem: “The larger leg of a right triangle is 3cm longer than its smaller leg” – this means the larger leg can be written as π + π and the smaller leg can be written as π. Also, since we’re told “the hypotenuse is 6cm longer than the smaller leg,” the hypotenuse can be written as π + π. π₯ π₯+6 π₯+3 Plugging these lengths into the Pythagorean Theorem, we get: π₯ 2 + (π₯ + 3)2 = (π₯ + 6)2 Now, we simplify and solve for x: π₯ 2 + (π₯ + 3)(π₯ + 3) = (π₯ + 6)(π₯ + 6) π₯ 2 + π₯ 2 + 3π₯ + 3π₯ + 9 = π₯ 2 + 6π₯ + 6π₯ + 36 2x 2 + 6π₯ + 9 = π₯ 2 + 12π₯ + 36 π₯ 2 − 6π₯ − 27 = 0 This factors to: (π₯ − 9)(π₯ + 3) = 0 Setting each equal to 0 and solving for x, we get: π₯−9=0 π₯=9 π₯+3=0 π₯ = −3 Because a triangle can’t have a side with a negative length, π₯ = 9 is the only answer that works. If we had been asked to solve for the other sides, we would simply add 3 and 6 to 9 to find the lengths. 14. Katie and Jennifer are playing a game. ο· ο· ο· Katie and Jennifer each started with 100 points. At the end of each turn, Katie’s points doubled. At the end of each turn, Jennifer’s points increased by 200. At the start of which turn will Katie first have more points than Jennifer? Be sure to read these types of problems VERY carefully. A good way to approach this problem is by creating a table: End of 1st turn: 200 300 Start: Katie Jennifer 100 100 End of 2nd turn: 400 500 End of 3rd turn: 800 700 Notice that Katie first has more points than Jennifer after the 3rd turn. BUT, the problem asks “At the start of which turn will Katie first have more points than Jennifer?” At the start of the 3rd turn Katie has 400 and Jennifer has 500. This means it is not until the start of the fourth turn that Katie has more points than Jennifer. 15. Alex walked 1 mile in 15 minutes. Sally walked 3,520 yards in 24 minutes. In miles per hour, how much faster did Sally walk than Alex? (Note: 1 mile = 1,760 yards.) To solve this problem we have to convert yards to miles. By dividing 3,520 by 1,760, we find that 3,520 is equal to two miles. This means that Sally walked 2 miles in 24 minutes. It also means that she walked 1 mile in half that amount of time (12 minutes). In miles per hour, Alex walked 4mph. We know this because if Alex can walk 1 mile in 15 min., he can walk 4 miles in 60 min. (60 min./15 min. = 4). In miles per hour, Sally walked 5mph. We know this because if Sally can walk 1 mile in 12 min., she can walk 5 miles in 60 min. (60 min./12 min. = 5). Sally walks 1mph faster than Alex. End of Part 1. Part 2 Calculators are allowed on Part 2. 3 1. Which expression is equivalent to √8π₯ 2 π¦ 3 π§ 4 ? Here we need to recall a few basic properties of exponents and radicals: 1) The small number outside of a radical, like the 3 above, is known as an index. It tells 3 us what root to look for. For example, √8 is equal to 2, because 2 β 2 β 2 = 8. Even though it doesn’t appear, a radical expression without an index, like √16, has an index of 2. This means it is asking for the square root of 16. The answer is 4, because 4 β 4 = 16. 2) Also recall that radical expressions can be written as exponents, and vice versa. 2 3 √(10π₯)2, for example, can be written as 10π₯ 3 . Notice that the index becomes the denominator and the power of the radicand (the number inside the radical) becomes the numerator. First, it may be helpful to break the expression down into smaller parts: 3 3 3 √π₯ 2 √8 √π¦ 3 3 √π§ 4 Now we can convert each part, simplify wherever possible: 3 √8 = π 3 π 3 3 √π¦ 3 = π¦ 3 = π¦ 1 = π √π₯ 2 = ππ 2 4 To finish, we put it all back together: 2π₯ 3 π¦π§ 3 3 3 2 4 a) 2π₯ 2 π¦π§ 4 b) 2π₯ 3 π¦π§ 3 c) d) 2π§ π₯ 2π₯ π§ 2. A school purchases boxes of candy bars. ο· ο· Each box contains 50 candy bars. Each box costs $30. 3 π √ π§ 4 = ππ How much does the school have to charge for each candy bar to make a profit of $10 per box? a) $0.40 b) $0.50 c) $0.80 d) $1.25 3. Energy and mass are related by the formula πΈ = ππ 2 . ο· ο· m is the mass of the object. c is the speed of light. Which equation finds m, given E and c? Expect at least one of these type of problems, where your given an equation and asked to solve for a different variable. This problem is an easier one; it can be solved in one step. To find m, which is the same thing as isolating m, all we have to do is divide both sides of the equation by π 2 , leaving us with answer choice d below. a) π = πΈ − π 2 b) π = πΈπ 2 c) π = π2 πΈ πΈ d) π = π 2 4. Suppose that the equation π = 20.8π₯ 2 − 458.3π₯ + 3,500 represents the value of a car from 1964 to 2002. What year did the car have the least value? (π₯ = 0 in 1964) The second you see a function with a degree of 2, you know you’re dealing with a quadratic function, which means you also know its graph is a parabola. Here we’re told the function (and therefore its graph) models the value of a car over a given period of time. Since this is part 2 of the EOC, we can use our calculators to solve. By hitting “y =” and entering the function exactly as it appears, the calculator will do the work of generating a table of values for us. All we have to do after entering the function is hit the “2nd” key and then “graph.” This brings up the table for the function. By scrolling up and down, we find that the least value for y is 975.5, which occurs when π₯ = 11. Don’t forget that when π₯ = 0 the year is 1964. This means that when π₯ = 1 the year is 1965, and so on. So when π₯ = 11, the year is 1975. Another option for solving this problem is to solve for the vertex of the parabola, which we know is the minimum point of this function because the function is positive (remember: positive ο smiley face). To solve for the vertex, we first solve for the axis of symmetry (see problem 4 in Part 1) and then plug that value back into the function for x and solve for y. a) 1965 b) 1970 c) 1975 d) 1980 1 5. Which expression is equivalent to (π₯ 3 )−3? 1 3 First, we can rewrite π₯ 3 in radical form: √π₯ (see an explanation of this in problem 1 above). Another exponent property we’ve learned deals with negative exponents. When an exponent is negative, we flip the base (take the reciprocal, in other words) and raise the denominator (bottom of the fraction) to the positive value of the exponent. 1 3 So, for ( √π₯)−3 , we end up with: 3 ( √π₯)3 3 Since ( √π₯)3 is just π₯, the answer is choice b. a) √π₯ b) c) d) 1 π₯ 1 π₯9 1 π₯ 27 6. The table below shows the average weight of a type of plankton after several weeks. What is the average rate of change in weight of the plankton from week 8 to week 12? Begin by listing the change in weight from week to week: from week 8 to 9, the change is 0.03; from 9 to 10, 0.07; 10 to 11, 0.11; 11 to 12, 0.24. + To find the average rate of change, all we do is find the sum of these four and then divide by four. 0.03 + 0.07 + 0.11 + 0.24 = 0.45 0.45 ÷ 4 = 0.1125 a) 0.0265 ounce per week b) 0.0375 ounce per week c) 0.055 ounce per week d) 0.1125 ounce per week 7. Dennis compared the y-intercept of the graph of the function π(π₯) = 3π₯ + 5 to the yintercept of the graph of the linear function that includes the points in the table below. x -7 -5 -3 -1 g(x) 2 3 4 5 What is the difference when the y-intercept of f(x) is subtracted from the y-intercept of g(x)? In this problem, before we can solve for the difference in y-intercepts, we have to come up with the function that is represented by the table. Notice that as g(x), which is y, πππ π(π¦) increases by 1, x increases by 2. Because we know that slope is determined by ππ’π(π₯) , we 1 know that the slope is 2. Alternatively, we can plug any two points into the point-slope π¦ −π¦ formula (π = π₯2−π₯1 ) and solve for m. 2 1 1 To find the y-intercept, we can plug m (2)and one of the ordered pairs into slopeintercept form, π¦ = ππ₯ + π and solve for b. Using the first ordered pair (-7, 2), we get: 1 2 = (−7) + π. Solving for b, we find the π = 5.5. 2 Finally, subtracting 5 from 5.5, we get 0.5. a) −11.0 b) −9.3 c) 0.5 d) 5.5 8. Cell phone Company Y charges a $10 start-up fee plus $0.10 per minute, x. Cell phone Company Z charges $0.20 per minute, x, with no start-up fee. Which function represents the difference in cost between Company Y and Company Z? Notice that we can write two functions, one for Company Y and one for Company Z: π(π₯) = 0.10π₯ + 10 π(π₯) = 0.20π₯ To solve, all we have to do is subtract to find Company X from Company Z to find the difference: 0.10π₯ + 10 −0.20π₯ + 0 π(π₯) = −0.10π₯ + 10 a) π(π₯) = −0.10π₯ − 10 b) π(π₯) = −0.10π₯ + 10 c) π(π₯) = 10π₯ − 0.10 d) π(π₯) = 10π₯ + 0.10 9. Monica did an experiment to compare the methods of warming an object. The results are shown in the table below. Time (Hours) 0 1 2 3 4 5 Method 1 Temperature (°F) 0 5 11 15 19 25 Method 2 Temperature (°F) 1.5 3 6 12 24 48 Which statement best describes her results? a) The temperature using both methods changed at a constant rate. b) The temperature using both methods changed exponentially. c) The temperature using Method 2 changed at a constant rate. d) The temperature using Method 2 changed exponentially. 10. Mario compared the slope of the function graphed below to the slope of the linear 4 function that has an x-intercept of 3 and a y-intercept of −2. What is the slope of the function with the smaller slope? Begin by finding the slope of the graph. Pick two points that hit both an x grid line and a y grid line, such as (1,2) and (4, 3). We can easily plug these points into the formula for π¦ −π¦ point-slope (π = π₯2 −π₯1 ) and solve for m. Or, easier yet, count up from (1,2) for rise and 2 1 1 over to (4, 3) for run. Either way, we find that the slope is 3. 4 To find the slope of the function that has an x-intercept of 3 and a y-intercept of −2, we can use π¦2 − π¦1 = π(π₯2 − π₯1 ). Recall that x- and y-intercepts tell us points on a graph. 4 If a point intercepts (hits) the x-axis at 3, it has a y-value of zero. This means the point is 4 (3 , 0). Likewise, a y-intercept of −2 is the point (0, −2). (If you are unable to do this in your head, plot the intercepts on a graph to find the ordered pairs.) Now that we know two points, let’s plug them in to π¦2 − π¦1 = π(π₯2 − π₯1 ). 4 −2 − 0 = π (0 − 3) 4 −2 = π (− 3) 4 To finish solving for m, we have to divide both sides of the equation by − 3. Your calculator will do this for you, but you should remember that dividing by a fraction 3 6 3 requires us to multiply by the reciprocal, so: −2 β − 4 = 4 = 2 1 3 is the smaller of the two slopes. 2 = π(3) − 2 a) b) 1 5 1 3 c) 3 d) 5 11. The boiling point of water, T (measured in feet) is modeled by the function π(π) = −0.0018π + 212. In terms of altitude and temperature, which statement describes the meaning of the slope? This is a linear function. We can tell because it is in the form π¦ = ππ₯ + π. Since the slope of the function, m, is negative, we know that the boiling point will be decreasing, not increasing. This allows us to omit answer choices e and f. By inserting different altitudes for a, we can test to see what happens. For example, when we evaluate the function for 1,000 feet, we find that the boiling point is 210.2. When we increase the altitude to 2,000 feet, which is an increase of 1,000 feet just as described in each of the answer choices, we find that the boiling point becomes 208.4. The difference is 1.8 degrees, not 18. e) The boiling point increases by 18 degrees as the altitude increases by 1,000 feet. f) The boiling point increases by 1.8 degrees as the altitude increases by 1,000 feet. g) The boiling point decreases by 18 degrees as the altitude increases by 1,000 feet. h) The boiling point decreases by 1.8 degrees as the altitude increases by 1,000 feet. 12. A line segment has endpoints J(2, 4) and L(6, 8). The point K is the midpoint of line JL. What is the equation of a line perpendicular to line JL and passing through K? a) π¦ = −π₯ + 10 b) π¦ = −π₯ − 10 c) π¦ = π₯ + 2 d) π¦ = π₯ − 2 13. A triangle has vertices at (1,3), (2, -3), and (-1, -1). What is the approximate perimeter of the triangle? a) 10 b) 14 c) 15 d) 16 14. The table below shows the area of several states. State Connecticut Georgia Maryland Massachusetts New Hampshire New York North Carolina Pennsylvania Area (thousands of square miles) 6 59 12 11 9 54 54 46 Delaware has an area of 2,000 square miles. Which is true if Delaware is included in the data set? First, try to eliminate answer choices. If Delaware is added to the data set, the number 2 will be included, and the data set will include 9 states. Adding such a small number to the data set will not increase the mean (average); it will cause the mean to decrease. Likewise, the range, which is the difference between the largest and smallest numbers in the data set, will not decrease; it will increase from 53 to 57. a) The mean increases. b) The range decreases. c) The interquartile range decreases. d) The standard deviation increases. 15. A college surveyed 3,500 of its students to determine if the students preferred music, movies, or sports. The results of the survey are shown in the relative frequency table below. Freshmen Sophomores Juniors Seniors Music 0.06 0.09 0.10 0.08 Movies 0.10 0.05 0.06 0.09 Sports 0.09 0.10 0.08 0.10 How many more seniors than juniors were included in the survey? Notice that if we add up all of the frequencies in the table, we get 1. This is because each frequency represents a percentage. To solve for the actual number of students in any of the categories, all we have to do is multiply the frequency and the total number of students surveyed (3,500). So, to find the number of seniors included in the survey we add the following products: 3,500(0.08) + 3,500(0.09) + 3,500(0.10) = 280 + 315 + 350 = πππ To find the number of juniors included in the survey we add the following products: 3,500(0.10) + 3,500(0.06) + 3,500(0.08) = 350 + 210 + 280 = πππ The difference is 945 − 840 = 105. a) 70 b) 105 c) 140 d) 175 16. A book club has 200 members. Each member was asked whether he or she prefers fiction or non-fiction books. The results are shown in the relative frequency table shown below. Age 21-30 31-40 Total Fiction 0.32 0.38 0.70 Non-Fiction 0.11 0.19 0.30 Total 0.43 0.57 1.00 Which statement is true? Another relative frequency table, just like we saw in problem 15. This time, since the answer choices are just random statements about various aspects of the table, let’s take the time to convert the frequencies in the table to actual values. Remember, the frequencies are percentages, so all we have to do is multiply each frequency by 200 (the total number of members in the book club) to find the actual number in each category. Here’s what we come up with: Age 21-30 31-40 Total Fiction 64 76 140 Non-Fiction 22 38 60 Total 86 114 200 Now this is a much easier question to answer. D is the only true statement. a) 6 more 31-40 year olds than 21-30 year olds prefer fiction. b) 38 members are 31-40 and prefer fiction. c) 43 members are 21-30 years old. d) 140 members prefer fiction. 17. The table below shows the distance a car has traveled. What is the meaning of the slope of the linear model for the data? a) The car travels 5 miles every minute. b) The car travels 4 miles every minute. c) The car travels 4 miles every 5 minutes. d) The car travels 5 miles every 4 minutes. 1 18. The area of a trapezoid is found by using the formula π΄ = 2 β(π1 + π2 ), where A is the area, h is the height, and π1 and π2 are the lengths of the bases. What is area of the trapezoid shown below? 1 We’re given the formula: π΄ = 2 β(π1 + π2 ). To solve, we need to substitute the values in the diagram below into the equation: 1 π΄ = 2 (4)((π₯ − 3) + (π₯ + 7)) According to the order of operations (PEMDAS), we first need to simplify inside the parentheses: π΄= 1 (4)(2π₯ + 4) 2 Next, we could distribute the 4, but to make things easier, we can first multiply 4 and 1 2 to get 2 outside the parentheses. After distributing the 2, we get: π΄ = 4π₯ + 8 a) π΄ = 4π₯ + 2 b) π΄ = 4π₯ + 8 c) π΄ = 2π₯ 2 + 4π₯ − 21 d) π΄ = 2π₯ 2 + 4π₯ − 42 19. A company produces packs of pencils and pens. ο· ο· ο· ο· ο· The company produces at least 100 packs of pens each day but no more than 240. The company produces at least 70 packs of pencils each day but no more than 170. A total of less than 300 packs of pens and pencils are produced each day. Each pack of pens makes a profit of $1.25. Each pack of pencils makes a profit of $0.75. What is the maximum profit the company can make each day? a) $338.75 b) $344.25 c) $352.50 d) $427.50 20. John mixed cashews and almonds. ο· ο· ο· John brought 4 pounds of almonds for a total cost of $22. The cost per pound for cashews is 60% more than the cost per pound for almonds. John bought enough cashews that when he mixed them with almonds, the mixture had a value of $6.50 per pound. Approximately what percent of the mixture, by weight, was cashews? a) 20% b) 25% c) 30% d) 35% 21. Lucy and Barbara began saving money the same week. The table below shows the models for the amount of money Lucy and Barbara had saved after x weeks. After how many weeks will Lucy and Barbara have the same amount of money saved? Lucy’s and Barbara’s savings are both modeled by linear functions. One way to determine after how many weeks Lucy and Barbara will have saved the same amount of money would be to graph both functions. If we were to do so, we would find that the two graphs intercepted (crossed) each other. Finding this point of intercept is one way to find after how many weeks they each had the same amount saved. A quicker way to solve for how many weeks it will be when Lucy and Barbara have the same amount of money saved is to set the two functions equal to one another, like so: 10π₯ + 5 = 7.5π₯ + 25 Now all we have to do is solve for x, which represents the number of weeks: 10π₯ = 7.5π₯ + 20 2.5π₯ = 20 π₯=8 e) 1.1 weeks f) 1.7 weeks g) 8 weeks h) 12 weeks 22. Collin noticed that various combinations of nickels and dimes could add up to $0.65. ο· ο· Let x equal the number of nickels. Let y equal the number of dimes. What is the domain where y is a function of x and the total value is $0.65? The first thing you need to know in order to solve this problem is what a domain is. Domain refers to the set of all the possible x-values that work or make sense in a given function. (You should know, too, that range refers to all the possible y-values that work or make sense in a given function.) Knowing that domain deals with x-values, we know immediately that we are being asked a question about nickels since we’re told to let x equal the number of nickels. The key to solving this problem is recognizing that only an odd number of nickels will yield an odd amount of money. This means that, because we’re looking for combinations that have a total value of $0.65, the domain can only contain odd numbers. Notice that there is only one set below made up entirely of odd numbers (d). (Zero is an even number.) a) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} b) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} c) {0, 1, 3, 5, 7, 9, 11, 13} d) {1, 3, 5, 7, 9, 11, 13} 2 23. The value of an antique car is modeled by the function π(π₯) = 107,000(1.009)(3π₯) where x is the number of years since 2005. By what approximate percent rate is the value of the car increasing per year? To find a percent rate we need to know the value of the car in two consecutive years. Since x represents years, we can easily solve for this by solving the function with x set equal to 1 (2006) and then again with x set equal to 2 (2007). Easier still, we can enter the function into the calculator, using y=, and check the table to find the value of the car in consecutive years. In year 1, the car is valued at 107,641. In year 2, its value is 108,286. To find the percent increase, we simply have to divide 108,286 by 107,641. This gives us 1.00599, which means the value has increased by roughly 0.006. As a percent, this is equivalent to 0.6%. i) 0.04% j) 0.14% k) 0.60% l) 1.40% 24. The table below shows the cost of a pizza based on the number of toppings. Which function represents the cost of a pizza with n toppings? If we think of the table above as inputs and outputs, we can identify the function that represents the cost of a pizza with n toppings by simply plugging in one of the inputs and checking to see if we get the corresponding output. In answer choice a, when we plug in 1 as our input, we’ll know if the function is correct if we get 12 for an output. πΆ(1) = 12 + 1.5(1 − 1) πΆ(1) = 12 + 1.5(0) πΆ(1) = 12 + 0 πΆ(1) = 12 A word of caution: it’s always a good idea to check more than one input. Notice that plugging 1 into answer choice d’s function also give 12 as an output. If we check 2, however, a gives 13.5 and d gives 14. Only answer choice a works. a) πΆ(π) = 12 + 1.5(π − 1) b) πΆ(π) = 1.5π + 12) c) πΆ(π) = 12 + π d) πΆ(π) = 12π 25. Paul sells chocolate chip cookies and peanut butter cookies. ο· ο· ο· ο· ο· Baking a batch of chocolate chip cookies takes 1.75 cups of flour and 2 eggs. Baking a batch of peanut butter cookies takes 1.25 cups of flour and 1 egg. Paul has 10 cups of flour and 12 eggs. He makes $4 profit per batch of chocolate chip cookies. He makes $2 profit per batch of peanut butter cookies. How many batches of peanut butter cookies should Paul make to maximize his profit? Recognizing that Paul stands to make more money from chocolate chip cookies – he makes twice as much when using double the number of eggs while using less than half the amount of flour – we know he should make as many batches of chocolate chip cookies as possible to maximize his profit. Next, we have to find out the maximum number of batches of chocolate chip cookies Paul can make with the ingredients he has (10 cups of flour and 12 eggs). He has enough eggs to make six batches but only enough flour for 5. If he makes 5 batches of chocolate chip cookies, he has enough ingredients left over to make 1 batch of peanut butter cookies. He should make only one batch of peanut butter cookies. a) 1 b) 2 c) 5 d) 8 26. The sequence below shows the number of trees a nursery plants each year. 2, 8, 32, 128, … Which formula could be used to determine the number of trees the nursery will plant next year, NEXT, if the number of trees planted this year, NOW, is known? In math, the word “sequence” implies a pattern. Some sequences have a common difference, such as 2, 5, 8, 11, 14, where 3 is added to each successive term. Other sequences have a common ratio, such as 1, 5, 25, 125, where each successive term is multiplied by 5. The sequence in this problem has a common ratio; each successive term is multiplied by 4. Solve this problem by plugging in numbers from the series. For example, if we use 2 as NOW, 8 would be NEXT. Testing these numbers in each of the functions below, we find the a, c and d work. This means we have to test another set of numbers. Let’s try 8 for NOW and 32 for NEXT. Now, only a works. The answer is a. a) ππΈππ = 4 β πππ 1 b) ππΈππ = πππ 4 c) ππΈππ = 2 β πππ + 4 d) ππΈππ = πππ + 6 27. There were originally 4 trees in an orchard. Each year the owner planted the same number of tress. In the 29th year there were 178 trees in the orchard. Which function, t(n), can be used to determine the number of trees in the orchard in any year, n? If there were originally 4 trees in an orchard and in the 29 th year there were 178 trees, we know two ordered pairs: (0, 4) and (29, 178). Because we know two ordered pairs, π¦ −π¦ we can solve for the slope, m, using π = π₯2−π₯1: 2 π= 1 178 − 4 174 = =6 29 − 01 29 Notice that only one of the answer choices has a slope of 6: c. a) π‘(π) = b) π‘(π) = 178 29 178 29 π+4 π−4 c) π‘(π) = 6π + 4 d) π‘(π) = 29π − 4 28. The vertices of a quadrilateral EFGH are E(-7, 3), F(-4, 6), G(5, -3), and H(2, -6). What kind of a quadrilateral is EFGH? For a problem like this you’ll want to graph each of the ordered pairs on a sheet of graph paper. Doing so reveals a rectangle that is not a square (two of its sides are longer than the other two). a) trapezoid b) square c) rectangle that is not a square d) rhombus that is not a square 29. R is the midpoint of segment PS. Q is the midpoint of segment RS. P is located at (8, 10), and S is located at (12, -6). What are the coordinates of Q? a) (4, 2) b) (2, -8) c) (11, -2) d) (10, 2) 30. The sequence below shows the total number of days Francisco had used his gym membership at the end of weeks 1, 2, 3, and 4. 4, 9, 14, 19, … Assuming the pattern continued, which function could be used to find the total number of days Francisco had used his gym membership at the end of the week n? Another sequence problem like #26 above. This time, however, the sequence is arithmetic because we add instead of multiply to get from one term to the next. a) π(π) = π + 5 b) π(π) = 5π − 1 c) π(π) = 5π + 4 d) π(π) = π2 31. The volume of a sphere is 2,400 cubic centimeters. What is the approximate diameter of 4 the sphere? (Volume of a sphere = 3 ππ 3 ) Begin by writing the formula with 2,400 filled in for the volume: 2400 = 4 3 ππ 3 Since we are being asked for the diameter, all we have to do is find the length of the radius, r, and multiply it by 2. First, we can multiply both sides of the equation by 3 to get rid of the fraction on the right side: 4 (3)2400 = (3) ππ 3 3 7200 = 4ππ 3 Next, we can divide both sides by 4, which leaves us with: 1800 = ππ 3 To divide both sides by pi (π), enter 1800 ÷ π into your calculator by hitting the “2nd“ key and then π (this is the key above the ÷ key). This gives us: 572.9578 = π 3 3 Finish solving for r by finding the cube root of each side, (MATH ο 4 will bring up √ . ) π = 8.30566 The diameter is twice this length: π = 16.6 a) 16.6 cm b) 10.1 cm c) 8.3 cm d) 4.2 cm 32. The number of points scored by a basketball player in the first eight games of a season are shown below. 15, 35, 18, 30, 25, 21, 32, 16 What would happen to the data distribution if she scored 24, 22, 27, and 28 points in her next four games? a) The data distribution would be less peaked and more widely spread. b) The data distribution would be less peaked and less widely spread. c) The data distribution would become more peaked and less widely spread. d) The data distribution would be more peaked and more widely spread. 33. The table below shows the shoe size and age of 7 boys. Name Tyrone Marcel Patrick Bobby Dylan Mike Jonathan Shoe Size (x) 6 6 7 8 9 10 12 Age (y) 9 11 15 11 15 16 17 Approximately what percent of the boys’ ages is more than 1 year different from the age predicted by the line of best fit for the data? a) 14% b) 29% c) 43% d) 57% 34. The scatter plot below shows the number of arithmetic errors 10 students made on a quiz and the amount of time the students took to complete the quiz. Which describes the relationship between the number of arithmetic errors the students made and the amount of time the students took to complete the quiz? If you remember back to the scatter plot project we did earlier in the year, we saw that a strong correlation between variables has a graph with points falling very close to a line, much like the graph above. The closer the point come to making a perfect line, the stronger the relationship. Likewise, the more scattered the points are, the weaker the relationship. Here, we have a very strong relationship between the variable, which is also negative because the slope of the line is negative. a) There is a strong positive relationship between the variables. b) There is a strong negative relationship between the variables. c) There is a weak positive relationship between the variables. d) There is a weak negative relationship between the variables. 35. An elevator can hold a maximum of 1,500 pounds. Eight people need to use the elevator. Bill had some measures from the data set of how much each person weighed. Which measure would be most useful to determine if the people can safely use the elevator? First, recall the definition of each of these terms. - The mean of a set of numbers is the same as the average. The median, like a median in the middle of a road, is the number that falls in the middle of the set of numbers. The mode is the number that occurs most frequently in a set of numbers. The interquartile range is the difference between the first quartile and third quartile of a set of numbers. An example of the interquartile range for the set of numbers 2, 5, 6, 9, 12 would be: Minimum = 2 First quartile = 3.5 Median = 6 Third quartile = 10.5 Maximum = 12 Thus the IQR = 10.5 – 3.5 = 7. The median and mode, as well as the interquartile range, are not as reliable as the mean for this particular scenario. Knowing the median weight tells us very little about the other people’s weight on the elevator. Likewise, the mode, only tells us if two or more people have the same weight. The mean (average weight of the people on the elevator) is by far the most useful. a) mean b) median c) mode d) interquartile range