San Diego Junior High Math Field Day 2011
Math Wits - 8
1. If 2011 is a base 3 number, what is its base 10 equivalent?
2. Grace is decorating her store window for a sale. She wants to design some art that looks like the design on the right. If she plans to paint the shaded area, how many square meters will she need to paint? Round to the nearest hundredth square meter. /3. If Herman’s test and quiz grade are equally weighted, his average would be 85%. If the test counts three times as much as the quiz, then his average would be 83% What percent grade did he receive on the test?
4. With an 80 kilometer per hour headwind, a plane can fly a certain distance in four hours. Flying in the opposite direction, with the same wind blowing, it can fly that same distance in one hour less. What is the plane’s air speed, in kph?
5. In a certain city, a birth occurs on the average every twenty-four minutes, and a death every half hour. A resident moves out of the city every one-and-a-half hours and a new person moves into the city every four-and-a-half hours. How many hours does it take, on the average, for the population to increase by one person?
6. What is the distance between (–3, 7) and (12, –5)? Express your answer in simplest radical form.
7. What is the measure, in degrees, of the smaller angle formed by the hands of a clock at 5:42?
8. Shawna plans to use sixty inches of wire to build the frames of two cubes, one with side lengths that are one-third as long as the other cube’s side length. What is the total volume of the two cubes, to the nearest hundredth of a cubic inch?
9. The sum of two numbers is 24. When each is squared, the difference is 144. What is the positive difference of the two numbers?
10. Find the area, in square centimeters, of a regular 10-sided polygon inscribed in a circle of radius 50 cm.
11. The average of five numbers is 66. If one of the five numbers is removed, the average of the four remaining numbers is 77. What is the value of the number that was removed?
12. Solve:
13. At a water pollution control facility, one pipe can fill a purification tank in four hours. Another pipe can fill the tank in six hours. How many hours will it take to fill the tank if both pipes are working together?
14. What is the only three-digit positive integer that has exactly five distinct, positive factors?
15. In the trapezoid ABCD, is parallel to , AB = CD, and point P lies on . What is the ratio of the area of triangle DPC to the area of trapezoid ABCD? Express your answer as a common fraction.
16. A survey of 1000 people found that 70% have a CD player, 85% have a telephone, and 45.2% have a computer. At least how many people have all three devices?
17. How many committees of six can be chosen from a management group of ten people if two of the members of the group cannot serve on the same committee?
18. Write an expression for the perimeter of a rectangle whose area is 6x2 +17x – 14.
19. What is the exact area of the circle that passes through the four unit squares, touching the midpoints of the sides, as shown on the right? /20. It is said that Diaphantus passed one-twelfth of his life in infancy and one-sixth more in youth. Then he married and spent one-seventh of his life and five more years. Then he had a son whom he survived by four years. The son reached only one-half of the father’s age. How many years old was Diophantus when he died?
21. A square with side length s is inscribed as shown in an equilateral triangle with side length t. What is the ratio t:s? Express your answer in simplest radical form. /22. A square and a regular hexagon have the same perimeter. What is the ratio of the area of the square to the area of the hexagon? Express your answer as a common fraction in simplest terms.
23. Maria has 5-pound weights and 10-pound weights. The total weight of the 5-pound set is the same as the total weight of the 10-pound set. If the whole set has 12 weights in all, what is their total weight, in pounds?
24. Pedro makes a 14-inch diameter pizza, which includes a 1-inch wide crust. Soren wants to make a 10-inch diameter pizza with the same percent of crust as Pedro’s pizza. To the nearest tenth of an inch, approximately how wide should the crust on Soren’s pizza be?
25. A restaurant mixes two gallons of milk containing 1% fat and three gallons of milk containing 2% fat. What is the percent of fat in the mixture, expressed to the nearest tenth?
26. A rectangle with length 16 cm and width 12 cm is inscribed in a circle. Find the exact area, in square centimeters, of the region inside the circle, but outside the rectangle.
27. A rectangular prism that is 4 inches by 5 inches by 12 inches is resting on a table on one of the faces that is smallest in area. It is filled with 8 inches of water. If it is then turned on its face that measure 4 by 12, how high, in inches, will the water reach from the bottom of the prism? Round to the nearest tenth of an inch if necessary.
28. Find the exact area, in square centimeters, of a hexagon with an apothem of four centimeters.
29. For what value of k will the three lines y = , x – 2y = 12, and y = kx + 84 intersect at a single point?
30. Simplify the following expression: .
State your answer as a common fraction.
31. The Pizza Pan restaurant charges $9.99 for a pizza that is 12-inches in diameter. Using the same pricing scheme, how much should it charge for a pizza that is 16 inches in diameter if the cost is proportional to the area of the pizza? Round to the nearest cent.?
32. When 10 cubic feet of water is poured into an empty fish tank that is a rectangular prism, the tank becomes 3/4 full. If the tank is 2 feet 8 inches wide and 40 inches long, how tall is the tank in feet and inches?
33. Find the y-intercept of a line that passes through the midpoint of a segment with endpoints (-5, 6) and (3, -8) and is perpendicular to that segment. Give your answer as a simple fraction.
34. Robert has three boxes of letters. The first box contains the six letters in the word “bubble,” the second box has the three letters in the word “zoo,” and the third contains the four letters in the word “taxi.” If he randomly selects one letter from each of the three boxes, what is the probability that the three letters will spell the word “box”?
35. Marcus takes eight steps every five seconds. With each step, he travels 17 inches. At this rate, how long will it take him to travel 1 mile? Round your answer to the nearest second.
36. If the length and width of a rectangular solid are each decreased by 20%, by what percent must the height be increased for the volume to remain the same? Give your answer to the nearest whole percent.
37. Suppose two eggs with bacon with bacon cost $2.70. One egg with bacon costs $1.80. At these rates, what should bacon alone cost?
38. Find k if 8x2 + 10x – 25 = (2x + k)(4x – k)
39. In the 1980s, the growth rate of the population of Honduras was about 2.7% per year. The 1993 population of Honduras was estimated at 5,170,000. Based on the 1980s growth rate, how many people (to the nearest thousand) will there be in Honduras in the year 2000?
40. Woody Woodpecker pecked at a 17 meter wooden pole until it cracked and the upper part fell, with the top hitting the ground 10 meters from the foot of the pole. Since the upper part had not completely broken off, Woody pecked away where the pole had cracked. How far above the ground was Woody? Round your answer to the nearest tenth of a meter.
41. A dart is thrown at a board 12 meters long and 5 meters wide. Attached to the board are 30 circular balloons, each with radius 10 cm. Assuming each balloon lies entirely on the board. Find the exact probability that a dart that hits the board will also kit the balloon.
42. The four interior angles of a quadrilateral are in the ratio 2:4:4:5. In degrees, what is the measure of the smallest interior angle of the quadrilateral?
43. Line m contains the points (–1, 3) and (1, –1). Line n is perpendicular to line m and contains the points (2, 2) and (–2, y). What is the value of y?
44. Two lines y = 2x –13 and 3x + y = 92 intersect. What is the value of x at the point of intersection?
45. Two of the four interior angles of a particular parallelogram are 130 degrees each. What is the number of degrees in each of the other two interior angles?
46. On an 8 unit by 15 unit rectangle, as shown at the right, congruent triangles ABC and A’B’C’ are constructed as illustrated, with corresponding sides parallel. If BC = 9 units, find the distance from A to A’. /47. How many positive integer factors of 22 x 32 x 5 are multiples of 12?
48. A can with 40 marbles in it weighed 135 grams. The same can with 20 marbles weighed 75 grams. What is the mass, in grams, of the can?
49. Find the value of three cubed, to the one half power. Write your answer in simplest radical form.
50. How many integers satisfy (n + 3)(n – 7) ≤ 0?
51. A square is inscribed in a circle. If the area of the square is 196 cm2, what is the exact area of the circle? /52. Solve for x: 25x2 + 49 = 70x.
53. The points A(0, 0), B(10, 0), C(10 + , –10 – ) and D(0, –10) form quadrilateral ABCD. What is the number of square units in the area of the quadrilateral? Express your answer in simplest radical form.
54. What is the area, in square centimeters, of a rhombus that has diagonals measure 19 cm and 22 cm?
55. For what value of n is 5 x 8 x 2 x n = 7! ?
56. The diameter of circle B is 16 cm. Angle ABC measures 55 degrees. What is the area of the sector ABC, to the nearest centimeter? /57. Evaluate for x = 6, y ¹ 0.
58. The length of the diameter of this spherical ball is equal to the height of the box in which it is placed. The box is a cube and has an edge length of 30 cm. How many cubic centimeters of the box are not occupied by the solid sphere? Express your answer in terms of p. /59. On the Moon, you toss a tennis ball from a height of 72 feet with an initial upward velocity of 24 feet per second. How many seconds will it take the tennis ball to reach the ground? Round to the nearest tenth. Use the model for vertical motion on the moon:
h = t2 – vt.
60. The four horsemen of the Apocalypse are trying to unlock the end of the world by determining the prime factorization of 5212011. Each horseman represents one of the prime factors. You can prevent the end of the world by finding the sum of the prime factors of 5212011. What is the sum of the prime factors?
THE END
NAME SCHOOL
2. / 22. / 42.
3. / 23. / 43.
4. / 24. / 44.
5. / 25. / 45.
6. / 26. / 46.
7. / 27. / 47.
8. / 28. / 48.
9. / 29. / 49.
10. / 30. / 50.
11. / 31. / 51.
12. / 32. / 52.
13. / 33. / 53.
14. / 34. / 54.
15. / 35. / 55.
16. / 36. / 56.
17. / 37. / 57.
18. / 38. / 58.
19. / 39. / 59.
20. / 40. / 60.
Score:
KEY
1. 58 / 21. or / 41.2. 2.36 (m2) / 22. / 42. 48 (degrees)
3. 81 (%) / 23. 80 (pounds) / 43. 0
4. 560 (kph) / 24. .7 (inches) / 44. 21
5. 18 (hours) / 25. 1.6 (%) / 45. 50 (degrees)
6. / 26. 100p – 192 (cm) / 46. 10 (units)