Practice C

Student Practice Questions A ferry boat shuttles tourists to an island every hour starting at 9 AM until its last trip starting at 3 PM. One day the boat captain notes that on the 9 AM trip there were 100 tourists on the ferry boat, and that on each successive trip, the number of tourists was 1 fewer than on the previous trip. How many tourists did the ferry take to the .island that day? (A) 585 (B) 594 (C) 672 (D) 679 (E) 694 "The ferry b'oat makes 7 trips to the island everyday." Solution Answer (D): The ferry boat makes 7 trips to the island. The number of tourists shuttled was 100 + (100 - 1) + (100 - 2) + (100 - 3) + (100 - 4) + (100 - 5) + (100 - 6) = 7 • 100 - (1 + 2 + 3 + 4 + 5 + 6) = 700 - 21 = 679. Point A = ( 4, 3) is rotated counterclockwise 90 ° around the or.igin to point A'. The point A' is reflected across the x-axis to poi_nt A". What is the distance between A ·and A"? (A) v'2 (B) 8 (C) 7v'2 (E) 14 (D) 10 "Vi�ualize the transformations by drawing." Solution Answer (C): The figure shows that A' The distance between A and A" is ✓(4- (-3)) 2 = (-3, 4) and A" + (3- (-4))2 = ✓72 + 72 = A' (-3,4) -----i I A"• (-3,-4) A (4,3) = 7v'2. (-3, -4). In a certain town bicycle license plates are to have 3 different letters, each letter chosen from A through F, followed by 3 different digits, each digit chosen from 1 through 6. The letters will appear in alphabetical order and the digits in numerical order. For example, ACE236 is allowed and ADB124 is not allowed. How many such license plates are possible? (A) 36 . (B) 72 ••Ther e ar e (�) from." (C) 120 (D) 240 (E) 400 = 20 letter combinations to choose Solution Answer (E): 1here are 6 6•5•4 ( ) =--=20 3 3•2•1 letter combinations to choose from, and each particular choice results in exactly one possible configuration for the letters in a license plate. Similarly, there are (�) = 20 possible configurations for the digits. Hence there are 20 • 20 = 400 possible license plates. Three children have a foot race in which the leader after 1 minute is declared the winner. Each child runs at a constant speed. At the end of the minute, Alicia has run 720 feet. Ben, who started 40 feet in front of Alicia, has ru_n 600 feet from his starting point. Cheryl, who started 40 feet in front of Ben, has run 400 feet from her starting point. For how many seconds during the race was Ben in the lead? (A) 8 (B) 10 (C) 12 (D) 15 (E) 20 "Use inequalities to represent the relationships." Solution Answer (A): The distance of Alicia, Ben, and Cheryl from Alicia's starting point after t minutes are, respectively, 720t, 40 + 600t, and 80 + 400t feet. For Ben to be in the lead we must have 40 + 600t > 720t and 40 + 600t > 80 + 400t. These inequalities are equivalent to 40 > 120t or and 200l > 40 ..... ->t 3 and Therefore Ben is in the lead when½ > t > ½, a time interval of½ - ½ , minutes, or 125 • 60 = 8 seconds. = 2 15 Suppose that ABCD is a square, M isthe midpoint of AB, and E is the intersection of CM and ED. What is:!? (A) 1 •• (B) � 4 (C) 2v'IO 5 (D ) v15 3 (E) i3 � ·-------- --.. Let F be the foot of the altitude to AB from E." Solution Let F be the foot of the altitude to AB from E and Answer (C): let EF = x. Then EB = ,v2x because 6BEF is a 45-45-90° By the triangle. Triangle EMF is similar to triangle CMB, so FM= Pythagorean Theorem applied to 6EMF, t Thus EB ,v2x ----------------,s��x 2 2,v2 2v'IO = vS�-�......--------------- F M Difficulty: Hard NCTM Standard: Geometry Standard for Grades 9-12: analyze properties and determine attributes of two- and three-dimensional objecjs. 27 How many integers in the set {1, 2, 3, ... , 15} cannot be the degree measure of the exterior angle of a regular polygon? ( A) 4 (B) 5 (C) 6 (D) 7 (E) 8 "The sum of the degree measure of the exterior angles of a convex p olygon is 360 . " Solution Answer (A): The sum of the degree measure of the exterior angles of a convex polygon is 360, so the degree measure of each exterior angle of a regular n-gon is 3�0 for n � --:t.' This is a positive integer k precisely when k is a factor of 360 and k � 120. For k � 15, all values are factors of 360 except 7, 11, 13, and 14. Let f be a ·function such that f(x - y) = f(x) + y for all real. numbers x ·and y and f ( 40) = 20. What is f (30)? (A) 10 11 (B) 20 (C) 30 (0) 40 (E) 50 f (30) = !(40 - 10) ." Solution Answer (C):, Note that !(30) = !(40 - 10) = !(40) + 10 = 20 + 10 = 30. The function f(z) = -z + 60 satisfies the given conditions. A and B be two independent events such that the probability that A occurs but B does not is¼, and the probability that B occurs but A does not is 1�. There Let p and q, for the probability that A and B both �ccur. What is p + q? are two possible values, (B) _220 (C) .!.!_ 20 13 (D) 20 (E) �4 "Let c be the probability that A and B both occur." Solution A and B both occur. Then the probability of A is c+ ¼, and the probability of Bis c+ 1�. Because A and Rare independent, r, = (c+ ¼)(c+ /0), so c2 - ��c+ }0 = 0. The two roots of this quadratic polynomial sum to the negative of the coefficient 0f C, W h•IC h •IS 13 20, Note: The two roots are c = 13 ±4'&129; they are both positive and less than Answer (D): Let c be the probabi lity that 3 4' In a right triangle, one altitude has length 60, and the sine of one �ngle is 0.6. What is the largest possible are� of this triangle? (A) 2250 (B) 2400 (C) 3000 (D) 3750 (E) 4250 .. Because the sine of one angle is 0.6, we have a 3x 4x - 5x right triangle." Solution Answer (D): Because the sine of one angle is 0.6, we have a 3x-4x-5x right triangle. The area of the triangle is times 60 times the length of the side corresponding to the altitude of length 60. To maximize the area this side should be the one measuring 5x. In this case the area of the triangle is both 3x • 4x and 60 • 5x. Therefore 6x2 = 150x, x = 25, and thus ½ ½• the area is ½• ½ • 60 • 125 = 3750.