ACT 2017
Name Date 1. The population of Green Valley, the largest suburb of Happyville, is 50% of the rest of the population of Happyville. The population of Green Valley is what percent of the entire population of Happyville? (A) 20% (B) 25% (C) 30% (D) 33 1 3 % (E) 50% 2. The sum, product, and average (arithmetic mean) of three integers are equal. If two of the integers are 0 and -5, the third integer is? (A) -5 (B) 0 (C) 2 (D) 5 (E) 10 3. If 7y = 2x 5, then x =? (A) 5y + 5 (B) 7 2 y 5 (C) 7 2 y + 5 (D) 7y 5 2 (E) 7y+5 2 2
4. A right circular cylinder has height and volume. What is the circumference of its base? (A) (B) (C) (D) (E) 5. Over all real numbers x, what is the maximum value of 4 sin 3x? (A) 1 (B) 2π 3 (C) 3 (D) 4 (E) 12 6. A particle travels 1 10 8 centimeters per second in a straight line for 4 10 6 seconds. How many centimeters has it traveled? (A) 2.5 10 2 (B) 2.5 10 13 (C) 4 10 2 (D) 4 10 14 (E) 4 10 48 3
7. Which of the following could be the equation of the function graphed in the above? (A) (B) (C) (D) (E) 8. For how many positive two-digit integers is the ones digit greater than twice the tens digit? (A) 16 (B) 20 (C) 28 (D) 32 (E) 36 4
9. What is the slope of the line 5y = 3x + 10? (A) -3 (B) - 3 5 (C) 5 3 (D) 2 (E) 10 1o. The bar graph above shows the number of people in attendance at each of the four meetings of the Maple Street Block Association that were held in 2011. Only members of the Block Association can attend the meetings, and no members joined or left the Block Association during 2011. Based on the bar graph, what is the least number of members the Maple Street Block Association could have had in 2011? (A) 61 (B) 65 (C) 67 (D) 72 (E) 268 5
11. A circle with center ( 3,4) is tangent to the x-axis in the standard (x,y) coordinate plane. What is the radius of this circle? (A) 3 (B) 4 (C) 5 (D) 9 (E) 16 12. In the figure above, O is the center of the circle and triangle A B O is equilateral. If the sides of triangle A B O are of length 6, what is the length of line B C? (A) 3 3 (B) 4 3 (C) 6 3 (D) 9 (E) 12 6
13. All numbers divisible by both 4 and 15 are also divisible by which of the following? (A) 6 (B) 8 (C) 18 (D) 24 (E) 45 14. A 25-foot ladder is placed against a vertical wall of a building, with the bottom of the ladder standing on concrete 7 feet from the base of the building. If the top of the ladder slips down 4 feet, then the bottom of the ladder will slide out (A) 4 feet (B) 5 feet (C) 6 feet (D) 7 feet (E) 8 feet 15. The triangle below is isosceles and is drawn to scale. What is the measure of N? (A) 22 (B) 68 (C) 78 (D) 79 (E) 89 7
16. Number of Cars produced in September The graph above shows the distribution of the number of cars produced in September by a group of employees of Company G. Based on the graph, what is the median number of cars produced in September for these employees? (A) 22 (B) 22.5 (C) 22.75 (D) 23 (E) 23.5 17. A circle with center (5,4) is tangent to the x-axis in the standard (x,y) coordinate plane. What is the radius of this circle? (A) 3 (B) 4 (C) 5 (D) 9 (E) 16 Number of Cars 8
18. For every 10 dollars Ken earns mowing lawns, he gives 3 dollars to his younger brother, Tim, who helps him. Which of the following gives the relationship between d, the number of dollars Ken earns, and t, the number of dollars he gives to Tim? (A) (B) (C) (D) (E) 19. In the figure below, X is on, XYZ measures 45, and AXZ measures 130. What is the measure of XZY? (A) 45 (B) 60 (C) 85 (D) 95 (E) 105 9
20. If Kelly buys pens priced at dollars each and pens priced at dollars each, which of the following expresses, in terms of and, the average (arithmetic mean) price, in dollars, of these pens? (A) (B) (C) (D) (E) 21. What is the maximum number of non-overlapping squares with sides of length 3 that will fit inside of a square with sides of length 6? (A) Two (B) Three (C) Four (D) Six (E) Nine 22. If a and b are any real numbers such that 0 < a < 1 < b, which of the following must be true of the value of ab? (A) 0 < ab < a (B) 0 < ab < 1 (C) a < ab < 1 (D) a < ab < b (E) b < ab 10
23. Of 5 employees, 3 are to be assigned an office and 2 are to be assigned a cubicle. If 3 of the employees are men and 2 are women, and if those assigned an office are to be chosen at random, what is the probability that the offices will be assigned to 2 of the men and 1 of the women? (A) 1 3 (B) 2 5 (C) 1 2 (D) 3 5 (E) 2 3 24. When graphed in the (x,y) coordinate plane, at what point do the lines x + y = 5 and y = 7 intersect? (A) ( 2,0) (B) ( 2,7) (C) (0,7) (D) (2,5) (E) (5,7) 25. The area of a trapezoid is 1 2 h(b 1 + b2), where h is the altitude, and b1 and b2 are the lengths of the parallel bases. If a trapezoid has an altitude of 5 inches, an area of 55 square inches, and one base 13 inches long, what is the length, in inches, of its other base? (A) 9.0 (B) 16.8 (C) 19.4 (D) 45.0 (E) 97.0 11
26. The circle shown above has center O and a radius of length 5. If the area of the shaded region is 20π, what is the value of x? (A) 18 (B) 36 (C) 45 (D) 54 (E) 72 27. The odometer of a new automobile functions improperly and registers only 2 miles for every 3 miles driven. If the odometer indicates 48 miles, how many miles has the automobile actually been driven? (A) 144 (B) 72 (C) 64 (D) 32 (E) 24 12
28. What integer most nearly approximate ( 50)( 80)? (A) 20 (B) 40 (C) 63 (D) 200 (E) 2,000 29. If, which of the following expresses in terms of? (A) (B) (C) (D) (E) 30. In the triangles above, 3(x-y) = (A) 15 (B) 30 (C) 45 (D) 60 (E) 105 13
31. Milk costs cents per half-gallon and cents per gallon. If a gallon of milk costs cents less than half-gallons, which of the following equations must be true? (A) (B) (C) (D) (E) 32. Over all real numbers x, what is the maximum value of 4 sin 3x? (A) 1 (B) (C) 3 (D) 4 (E) 12 33. A, B, C, and D are points on a line, with the midpoint of segment. The lengths of segments,, and are,, and, respectively. What is the length of segment? (A) (B) (C) (D) (E) 14
34. What is the largest possible product for 2 even integers whose sum is 34? (A) 64 (B) 68 (C) 120 (D) 240 (E) 288 35. A geologist has 10 rocks of equal weight. If 6 rocks and a 10-ounce weight balance on a scale with 4 rocks and a 22-ounce weight, what is the weight, in ounces, of one of these rocks? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8 36. A particle travels 1 10 8 centimeters per second in a straight line for 4 10 6 seconds. How many centimeters has it traveled? (A) 2.5 10 2 (B) 2.5 10 13 (C) 4 10 2 (D) 4 10 14 (E) 4 10 48 15
37. Four distinct lines lie in a plane, and exactly two of them are parallel. Which of the following could be the number of points where at least two of the lines intersect?. Three. Four. Five (A) only (B) only (C) and (D) and only only (E),, and 38. A woman drove to work at an average speed of miles per hour and returned along the same route at miles per hour. If her total traveling time was hour, what was the total number of miles in the round trip? (A) (B) (C) (D) (E) 16
39. The graph of y = is shown below. Among the following, which is the best representation of y =? (A) (B) (C (D) (E) 40. If 7y = 2x 5, then x =? (A) 5y + 5 (B) y 5 (C) y + 5 (D) (E) 17
41. It costs dollars to ship a package that weighs pounds if and only if. If Alan's package weighs pounds and costs dollars to ship, which of the following must be true? (A) (B) (C) (D) (E) 42. In a shipment of 1,000 light bulbs, of the bulbs were defective. What is the ratio of defective bulbs to non-defective bulbs? (A) (B) (C) (D) (E) 18
43. In the quadratic equation above, is a constant. The graph of the equation in the contains the points and. What is the value of? (A) (B) (C) (D) (E) 44. The function is graphed in the above. If the function is defined by, for how many values of between and does equal? (A) None (B) One (C) Two (D) Three (E) More than three 19
45. Which of the following is divisible by 3 (with no remainder)? (A) 2,725 (B) 4,210 (C) 4,482 (D) 6,203 (E) 8,105 46. On the last day of a one-week sale, customers numbered 149 through 201 were waited on. How many customers were waited on that day? (A) 51 (B) 52 (C) 53 (D) 152 (E) 153 47. In the figure below, and is 6 units long. What is the length of side AC? (A) 12.5 (B) 25 (C) (D) 50 (E) Cannot be determined from the given information. 20
48. The perimeter of a 7-sided figure is 15. If the length of each side of the figure is increased by 2 units, what is the perimeter of the new figure? (A) 17 (B) 22 (C) 24 (D) 29 (E) 30 49. Over all real numbers x, what is the maximum value of 4 sin 3x? (A) 1 (B) (C) 3 (D) 4 (E) 12 50. If, the value of can be which of the following?... (A) only (B) (C) only only (D) and only (E),, and 21
51. What is the largest possible integer value of n for which 5 n divides 50 7? (A) 2 (B) 7 (C) 9 (D) 10 (E) 14 52. What is the slope of the line 5y = 3x + 10? (A) 3 (B) 3 5 (C) 5 3 (D) 2 (E) 10 53. If, which of the following must be equivalent to? (A) (B) (C) (D) (E) 22
54. In the figure below, and is 10 units long. What is the area, in square inches, of ABC? (A) 12.5 (B) 25 (C) (D) 50 (E) Cannot be determined from the given information 55. The graph of is shown above. If, and if is on the graph of, which of the following must be true? (A) (B) (C) (D) (E) 23
56. A line segment containing the points (0, 0) and (12, 8) will also contain the point (A) (2, 3) (B) (2, 4) (C) (3, 2) (D) (3, 4) (E) (4, 2) 57. If is an odd integer, which of the following is an even integer? (A) (B) (C) (D) (E) 58. What are the values of a and b, if any, where a b 2 < 0? (A) a < 0 and b 2 (B) a < 0 and b = 2 (C) a 0 and b > 2 (D) a > 0 and b < 2 (E) There are no such values of a and b. 24
59. If, then (A) (B) (C) (D) (E) 25
Answer Key 1. D Let equal the population of Green Valley, and let equal the entire population of Happyville. Then is the population of the rest of Happyville. is of. 2. D You have three integers, 0,- 5, and x, whose sum, product, and average are equal. Since 0 is one of the numbers, the product is 0. So the sum and average are also equal to 0. Therefore, - 5 + x + 0 = 0, and x = 5. 3. E 7y = 2x 5, so 2x = 7y + 5, x = 7y+5 2. 4. C The volume of a cylinder with radius and height is equal to. Therefore, and, so the radius of the cylinder is. The circumference of a circle with radius is equal to ; therefore, the circumference of the base is equal to. 5. D The maximum value for sin X, where X is any function of x, is 1 so sin 3x 1 for all x. Then multiplying by 4, 4 sin 3x 4, so the maximum value of 4 sin 3x is 4. A, B, C. When x =, 4 sin 3x = 4 sin = 4 sin = 4 1 = 4. Since the values in F H are less than 4, none can be the maximum value of 4 sin 3x. E. If 12 were a value of 4 sin 3x, then 12 = 4 sin 3x for at least one value of x. But then 3 = sin 3x for at least one value of x. But sin 3x 1 for all x, so 12 can't be a value of 4 sin 3x. 26
6. C Using D = rt, D = = 4 10 8 6 = 4 10 2 cm 7. D The function with equation and the function with equation each have a minimum value of when, but the function graphed does not have a minimum value of, so these options cannot be correct. The graph of the function with equation contains the point, but the function graphed does not contain any points with negative, so this option cannot be correct. The graph of the function with equation is not symmetric with respect to the, so it cannot be the equation of the function graphed. Therefore, the only equation that could correspond to the function graphed is 8. A.. Its graph is the absolute value of a parabola opening upward with vertex at If the tens digit is 1, there are 7 positive two-digit integers with the ones digit greater than twice the tens digit: 13 comma 14 comma 15 comma 16 comma 17 comma 18 and 19. If the tens digit is 2, there are 5 positive two-digit integers with the ones digit greater than twice the tens digit: 25, 26, 27, 28 and 29. If the tens digit is 3, there are 3 positive two-digit integers with the ones digit greater than twice the tens digit: 37, 38 and 39. If the tens digit is 4, there is 1 positive twodigit integer with the ones digit greater than twice the tens digit:. If the tens digit is equal to or greater than 5, then the ones digits, which cannot be greater than 9, is not greater than twice the tens digit. Therefore, there is a total of 7 + 5 + 3 + 1 = 16 positive two-digit integers for which the ones digit is greater than twice the tens digit. 9. B 5y = 3x + 10 (5y) = ( 3x + 10) y = x + 2 This is the slope-intercept form, y = mx + b, where m is the slope, so the slope is - 3 5 27
10. D The highest attendance was at the spring meeting, which had people attending. Since only members can attend the meetings, these Street Block Association; thus, there must have been at least people must have been members of the Maple members of the Block Association. If the, and people who attended the winter, summer and autumn meetings, respectively, were all among the people who attended the spring meeting, then the Block Association could have had as few as members in all. Therefore, based on the bar graph, the least number of members the Maple Street Block Association could have had in 11. B is. If the circle with center ( 3,4) is tangent to the x-axis, the point of tangency is ( 3,0) and the distance between these points is 4, so the radius is 4. 12. C Since is equilateral, each of its angles has measure. It follows that has measure. Since and are radii of the same circle, they are of equal length, and so is isosceles. Hence and each have measure. Thus has measure, and so is a right triangle. Since, the side opposite the angle in, is of length, it follows that, the side opposite the angle, is of length. 13. A All numbers divisible by both 4 and 15 are the multiples of 4 and 15. Since 4 and 15 have no prime factor in common, then the least common multiple of 4 and 15 is equal to their product, namely 4 x 15 = 60. Every other multiple of 4 and 15 is divisible by 60. Thus, if 60 is divisible by a number then all the multiples of 4 and 15 are divisible by that number. Therefore, it is enough to check by which number in the given options is 60 divisible. Only 6 divides 60 and none of the other do. 28
14. E The ladder, the wall, and the ground form a right triangle with a 25-foot hypotenuse. At first, the bottom of the ladder is 7 feet from the base of the building, so one leg of the right triangle measures 7 feet; the length of the other leg, x, can be found by solving (7^2) + (x^2) = (25^2), which is the Pythagorean theorem. From this, you can figure out that the other leg measures 24 feet. After the ladder slips down 4 feet, the 24-foot leg of the right triangle becomes 20 feet long. The other leg then has to be 15 feet long. This length is found by solving (20^2) + (y^2) = (25^2), which is again the Pythagorean theorem. Since the distance between the bottom of the ladder and the base of the building increases from 7 feet to 15 feet, the amount that the bottom of the ladder slides out is 8 feet. 15. D From the figure,, so M N as base angles of an isosceles triangle. Then, 22 + 2m( N) = 180 ; m( N) = = 79. 16. A There are 5 + 6 + 5 + 8 + 6 + 1 equals 31 employees represented in the graph. If the employees are put in order according to the number of days spent producing cars from least to greatest (or from greatest to least), the median number of days spent producing cars is the number of days spent by the employee in the middle of the list. Since 31 divided by 2 equals 15.5, this is the 16th employee (there will be 15 employees before and 15 employees after the 16th employee in a list of 31 employees). From the graph, it can be seen that the 16th employee is in the group that spent 22 days producing cars (since 5 employees spent 20 days producing cars, 6 employees spent 21 days and 5 employees spent 22 days). Therefore, the median number of days spent producing cars in September for these employees is 22. 29
17. B If the circle with center (5,4) is tangent to the x-axis, the point of tangency is (5,0) and the distance between these points is 4, so the radius is 4. 18. B For every dollars Ken earns, he gives dollars to Tim. Thus if Ken earns dollars and gives Tim dollars, the proportion holds. Multiplying both sides of this proportion by gives. 19. C AXZ is an exterior angle of XZY. By the exterior angle theorem, 45 + m( XZY) = 130, so m( XZY) = 130 45 = 85. Or, by definition of supplementary angles, m( YXZ) + 130 = 180, so m( YXZ) = 180 130 = 50. Then for XZY, the angle sum is 45 + 50 + m( XZY), and 180 = 95 + m( XZY) and m( XZY) = 180 95 = 85. 20. D The average price of the pens is equal to the total price of all of the pens divided by the total number of pens. There are pens priced at dollars each, so the total price of these pens is dollars. There are pens priced at dollars each, so the total price of these pens is dollars. Thus the total price of all of the pens is dollars, and the total number of pens is, the average price, in dollars, of all of these pens is. 21. C A square with sides of length has area, and a square with sides of length has area Thus at most squares of side length can fit inside a square of side length without overlapping. And in fact, it is possible to fit the four squares of side length inside a square of side length with no overlap; if the four squares with sides of length are arranged in two rows with two squares in each row, they will fit inside of the square with sides of length 30
without overlapping. Therefore, the maximum number of non-overlapping squares with sides of length that will fit inside of a square with sides of length is four. 22. D If all members of 0 < a < 1 < b are multiplied by a, which is positive since 0 < a, the inequality is 0 < a 2 < a < ab, so a < ab. If all members of 0 < a < 1 < b are multiplied by b, which is also positive since 0 < b by transitivity, the inequality is 0 < ab < b < b 2, so ab < b. Combining gives a < ab < b. Let a =, b = 2. These are valid values since 0 < < 1 < 2. Then ab = 1. 23. D Of the 5 office workers, 3 are to be assigned an office. This is an example of combinations: to find the number of ways of choosing 3 of the 5 workers, you can count the number of ways of selecting the workers one at a time and then divide by the number of times each group of 3 workers will be repeated. There are 5 ways of choosing the first worker to get an office. Then there will be 4 ways of choosing the second worker to get an office, and 3 ways of choosing the third worker. This is a total of 5 times 4 times 3 = 60 possibilities. In these 60 possible selections, each distinct group of 3 workers will occur 3 times 2 times 1 = 6 times. (There are 3 possibilities for the first worker chosen from the group, 2 for the second worker chosen, and only 1 for the third.) Therefore, there are 60 over 6 = 10 different ways the 3 workers who get an office can be chosen from the 5 workers. How many of these 10 possible groups of 3 workers consist of 2 men and 1 woman? From the 3 male workers, 2 can be chosen in 3 different ways. There are 2 possibilities for the female worker. Therefore, 3 times 2 = 6 of the groups of 3 workers consist of 2 men and 1 woman. Since there are 10 different ways the 3 workers who get an office can be chosen, and 6 of these possible groups of 3 workers consist of 2 men and 1 woman, the probability that the offices will be assigned to 2 men and 1 woman is 6 over 10, or 3 over 5. 31
24. B If x + y = 5 and y = 7, then x + 7 = 5 and x = 2. The point of intersection is ( 2,7). Check: 2 + 7 = 5 and 7 = 7, so ( 2,7) is on both lines. A, C, D, E cannot be correct since two distinct lines intersect in at most one point, and ( 2,7) is the point of intersection, so none of the points in A, C, D, and E can be on both lines. 25. A Substituting into the formula, 55 = (5)(13 + b), so 110 = 5(13 + b), 110 = 65 + 5b, 5b = 45, b = 9. 26. A In order to find the value of, you should first determine the measure of the angle that is located at point in the right triangle. To determine this angle, you must calculate what fraction of the circle s area is unshaded. The radius of the circle is and its area is, or. The area of the shaded region is, so the area of the unshaded region must be. Therefore, the fraction of the circle s area that is unshaded is, or. A circle contains a total of degrees of arc, which means that of degrees, or degrees, is the measure of the angle at point in the unshaded region. Since you now know that two of the three angles in the triangle measure degrees and degrees and that the sum of the measures of the three angles is always degrees, the third angle must measure degrees. Therefore,. 27. B In this problem, you are told that the odometer registers only miles for every miles driven. So the ratio of miles registered to miles driven is to or. This can be expressed as 32
If the odometer indicates relationship as follows: miles, the actual miles can be found using using the above So if the odometer indicates miles, the actual number of miles driven is. The most important thing with ratios is to be consistent in the way you set them up. If you mix up the terms, you won t get the correct answer. For instance, if you put the registered mileage in the numerator of one ratio but the actual mileage in the numerator of the other ratio, you will come up with a wrong answer: 28. C or 60; 63 60. Or, is about 7, is about 9, so is about 63. So 63 is the best approximation. 29. C Since, it follows that. Therefore,, from which it follows that. 33
30. C To evaluate the expression, it is necessary to determine the values of and. Since the sum of the measures of the angles of any triangle is degrees, and a right angle has measure degrees, the value of can be found by solving the equation. This yields. The value of can be found by solving the equation. This yields. Finally, the values for and are substituted into the expression to obtain. 31. D The cost of a gallon of milk is equal to cents less than the cost of half-gallons. This can be written algebraically as. Subtracting from both sides of the equation yields. 32. D The maximum value for sin X, where X is any function of x, is 1 so sin 3x 1 for all x. Then multiplying by 4, 4 sin 3x 4, so the maximum value of 4 sin 3x is 4. 33. B Try to draw the figure. You might be tempted to locate point first. Unfortunately, you don't have enough information about, yet, to place it. You can place,, and because is the midpoint of. You know the lengths of three of the line segments: Because you know where is, you can label the length of. Because is the midpoint of, you know that and are each units long. 34
Where can you place point? It has to be units from, because. It also has to be units from, because. So the only location for is between and, but closer to. Place point and mark the distances. It is now an easy matter to figure out the answer to the question: is units. is units closer to than, so is units. 34. E If n is one integer, then 34 n is the other. To maximize n(34 n), consider y = x(34 x) = 34x x2 = (x2 34x) = (x2 34x + 289) + 289 = (x 17)2 + 289. The graph is a parabola that turns downward and has maximum point (17,289). Applying this to the task of maximizing n(34 n) for even integers n, the closest even value to 17 is 16 (or 18), so n = 16 and 34 n = 18 (or n = 18 and 34 n = 16). The product is 288. So the even integers are 16 and 18. Their sum is 34, and their product is maximum. 35
35. C Let equal the weight of each rock, in ounces. The weight on one side of the scale is ounces and the weight on the other side is ounces. Since the two weights balance, you can write the equation. Solving this equation yields. 36. C Using D = rt, D = = 4 10 8 6 = 4 10 2 cm 37. D Since exactly two of the four lines are parallel, the other two lines are not parallel. If the two non-parallel lines intersect at a point that is not on either one of the parallel lines, then the configuration of lines will give a total of points of intersection. (The best way to verify this is by drawing the two parallel lines and then putting in the other two lines.) If, on the other hand, the two non-parallel lines intersect at a point that is on one of the parallel lines, then there will be a total of points of intersection in the figure. (Again, a sketch is the best way to verify this.) Any arrangement of the four lines will again yield either can t obtain four points of intersection, the correct answer is and or points of intersection. Since you only. 38. C Let represent the time it took the woman to drive to work. Since her total traveling time was hour, the time it took her to return home is. The distance in miles can be found using the formula. The distance traveled to work is, and the return distance is. Since the distance to work is the same as the return distance,. Solving for yields. So the distance one way is miles. miles. The total number of miles in the round trip is, or 36
39. D When x > 7, = x 7, so the graph of y = will be identical to the graph of y = for all x > 7. When x < 7, = (x 7), so the graph of y = will be the reflection about the x- axis of the graph of y = for all x < 7. 40. E 7y = 2x 5, so 2x = 7y + 5, x =. 41. C If Alan's package weighs pounds and costs dollars to ship, then. Subtracting across these inequalities gives the equivalent, and so. By the definition of absolute value, it follows that must be true. 42. D of 1,000 light bulbs were defective, so (1,000) or 25 bulbs were defective. The rest, 1,000 25 or 975, were non-defective. The ratio of defective to non-defective is =. 43. A Since the graph of in the contains the point, it follows that substituting the value into yields. Hence., which simplifies to, or. Therefore, 37
Alternatively, since the graph of in the contains the points and, and the coefficient of is, the equation is equivalent to, which multiplies out to. Therefore,. 44. C The graph of the function is the same as the graph of the function translated up units. The function intersects the three times, but if the graph is shifted up units, then the graph will intersect the only twice. Therefore, is equal to for exactly two values of between -5 and 15. 45. C A rule for divisibility by 3 is: if the sum of the digits of a number is divisible by 3, then so is the number; 4 + 4 + 8 + 2 = 18, which is divisible by 3. 46. C The number of customers who were waited on that day is (201 minus 149) + 1 = 53, as the total number of customers is those customers numbered 149 to 201 inclusive. 47. E It is impossible to find side AC without more information, because there are infinitely many isosceles triangles having a base that is 10 units long. 48. D The perimeter is the sum of the lengths of all the sides. Since each of the seven sides is to be increased by 2, there will be a total increase of 14 in the perimeter. The original perimeter, 15, plus the increase, 14, gives the new perimeter, 29. 49. D The maximum value for sin X, where X is any function of x, is 1 so sin 3x 1 for all x. Then multiplying by 4, 4 sin 3x 4, so the maximum value of 4 sin 3x is 4. 38
50. D This is a very easy substitution to make: Can the value of be? Substitute for : If, then.. 51. E Since, it follows that. Applying the laws of exponents to the preceding equality yields. From this representation of, it follows that divides and that is the largest value of for which divides. 52. B 5y = 3x + 10 (5y) = ( 3x + 10) y = 3 5 x + 2 This is the slope-intercept form, y = mx + b, where m is the slope, so the slope is 3 5 39
53. E If, then, so 54. E It is impossible to find the area of the triangle without more information, because there are infinitely many isosceles triangles having a base that is 10 units long. 55. D The second coordinate of a point on the graph corresponds to. From the graph, you can see that for, the smallest value of is and the greatest value is, so. 56. C The slope of the line containing the points and is, or. If is another point on this line segment, then and must satisfy, or. Of the five choices, is the only point for which. 57. E If is an odd integer, then and are odd integers. Similarly, choices (C) and (D) are odd integers. Since an odd integer subtracted from another odd integer is always an even integer, is even. 58. A If a b 2 < 0, one of a or b 2 must be negative and one must be positive. Since b 2 can't be negative, a must be negative. Then b 2 must be positive, so b 2. 40
59. C You are given that. Dividing both sides of the equation by gives 41