Topical Problems -- WSMC State Contest -- 4/18/1998

Questions

Directions: Mark the correct response on the answer form. Make sure that the answer is legible and that your name(s) are on the answer form. If you are a team, also put your team number on the answer form. Remember the scoring: 5 points for a correct response, 1 point for no response, and 0 points for an incorrect response. If the exact answer is not one of the choices, then choose the answer closest to the exact answer.

A rectangular bathtub 21 inches by 52 inches is filled to a depth of 15 inches. It drains ar the rate of one cubic foot per minute into a septic tank which is 8 feet in diameter and 10 feet deep.

1. How many minutes does it take to drain the tub?

a) 5.4 b) 9.5 c) 11.2 d) 13.6

2. What is the rate (in./min.) at which the water level in the septic tank rises?

a) .25 b) .375 c) .5 d) .625

3. At that rate, how long (hrs.) would it take to fill the septic tank?

a) 5 b) 6 c) 7 d) 8

A polyhedron has as its vertices the midpoints of each edge of a rectangular prism.

4. The sum of the number of faces and the number of edges of the polyhedron is:

a) 46 b) 38 c) 36 d) 18

5. If the surface area of the prism doubles, each dimension increasing by the same factor, by what factor is the volume of the polyhedron increased?

a) 2 b) 2.8 c) 4 d) 8

6. The ratio of the surface area of the polyhedron to the rectangular prism is:

a) 1:2 b) 2:3 c) 3:4 d) 7:8

Seven candidates for a political office need to be seated in front of an audience to answer questions in an open forum before the election. Let the candidate names be A, B, C, D, E, F, and I.

7. How many different seating arrangements are possible?

a) 7 b) (7)(7) c) 77 d) 7!

8. If no two vowels and no two consonants can sit next to each other, then how many different seating arrangements are possible?

a) 144 b) 210 c) 840 d) 5040

A cat burglar in Seattle has been sighted in an apartment window 23' from the top of a building. The police know they must enter the window and the door to the apartment simultaneously to catch him. The only way to do it is to throw a hook from the top of another building, which is the same height, to catch on the window ledge. The buildings are 80' apart, and an extra 3' of rope must be allowed so that the rope can be secured.

9. To the nearest foot, how what is the shortest rope that can be used?

a) 85' b) 86' c) 87' d) 88'

10. The police realize that the noise from the hook will alert the burglar, and decide to throw the hook to the window one floor above, which is 9' higher. What would be the percent of decrease in the length of the rope?

a) 2.31% b) 2.34% c) 2.39% d) 2.41%

11. If the maximum angle of depression of the rope can be 15°, which of the windows could be used? (disregard the surprise factor)

a) upper only b) lower only c) neither one d) both

One plant is now 12 cm tall and will grow 2 cm per week. A second plant is now 3 cm tall and will grow 5 cm per week.

12. At the end of what week will the ratio of their heights be 2:1 for the second time?

a) 12 b) 21 c) 33 d) 54

13. In order for this relationship to exist, how many days earlier would one plant have to have been planted than the other?

a) 15 b) 22 c) 31 d) 38

Nick's Acres irrigates the blueberry field with a traveling irrigation system. The hose reel has an inside core of diameter 2 feet, endplates of diameter 4 feet, and an inside length of 4 feet. The hose is three-inches in diameter, and the full reel holds all but the last 4 feet of the hose, which remains connected to the sprinkler. Nick's field is 1090 feet long and 200 feet wide. The sprinkler waters in a circular pattern of radius 50 feet.

14. What is the length of the hose (feet) that is connected to the sprinkler?

a) 406 b) 503 c) 607 d) 703

15. If the entire field must be watered directly by the sprinkler (ie, no unwatered areas, and the assumption that no water runs around on the ground to water any gaps in the pattern), and the overlap of the sprinkle pattern must be at least 30 feet, how many settings must be made to water the entire field?

a) 28 b) 36 c) 48 d) 54

A patio is to be made of concrete. It is to be 12 feet wide (from the house to the outer edge), and 32 feet long. The ground depth slopes from 6.5 inches at the house wall to 10" at the outer edge. The concrete is to be level with the house, only 3 inches deep, with a one-half inch slope from the house wall to the outer edge,so that means a sand fill will have to be made before the concrete can be poured. Sand costs $6 per cubic yard, and concrete costs $25 per cubic yard. You may only order an integral number of yards of either type of material.

16. How many cubic yards of sand fill will be needed?

a) 5 b) 6 c) 7 d) 8

17. How many cubic yards of concrete will be needed?

a) 4 b) 5 c) 6 d) 7

18. What will be the total cost of the sand and concrete?

a) $124 b) $136 c) $144 d) $156

A train engine has a maximum speed of 120 km/h. With cars attached, its maximum speed is diminished by an amount proportional to the square root of the number of cars. With four cars attached, its maximum speed is 90 km/h.

19. What is the train's maximum speed with 16 cars attached?

a) 60 b) 45 c) 30 d) 0

20. The largest number of cars it can move is:

a) 11 b) 12 c) 36 d) 64

Consider a sequence which has the sum formula Sn = 2 + n3n+1 where n is the number of terms in the sum, that is, Sn = a1 + a2 + a3 + ... + an and n> 1.

21. What is the value of the fourth term in the sequence?

a) 1130 b) 2514 c) 2413 d) 9

22. What type of sequence is this?

a) arithmetic b) geometric c) Fibonacci d) none of a,b,or c

Careless Dare is a free-fall parachutist. She is planning to jump from 10,000 feet. She knows that she will fall 48 feet in the first two seconds, and 64 more feet in the next two seconds. Assume that the total effect of all forces results in her path being a parabola generated by the given data.

23. How many feet will she have fallen in the first eight seconds?

a) 96 b) 128 c) 288 d) 1024

24. If she plans to deploy her parachute at 3,000 feet, how long (sec.) will she have for her free-fall?

a) 20.9 b) 33.3 c) 41.8 d) 54.3

25. If it takes five seconds for her parachute to deploy and slow her down enough for a safe landing, what is the lowest altitude at which she can safely pull the ripcord to begin deployment?

a) 1370 ft b) 2298 ft c) 3240 ft d) 3600 ft

A rectangular prism has dimensions 2 x 4 x 8.

26. If an ant were to crawl on the surface of the prism from any vertex to its opposite vertex (ie. the vertex at the other end of the internal diagonal), the shortest distance it could crawl would be:

a) 2 + ÷`80 b) 4 + ÷`68 c) 8 + ÷`20 d) 10

27. If the ant were to crawl from any vertex to its opposite vertex, crawling only on edges and not revisiting a vertex or edge enroute, how many unique paths would be possible?

a) 12 b) 14 c) 16 d) 18

In a college department, students labeled1 through 8 need to take final exams in classes labeled A through H. The following matrix describes which student takes which courses. A 0 means that the student does not take the course, and a 1 means that the student does take the course.

Course

Student	A B C D E F G H
1 2 3 4 5 6 7 8	1 0 1 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 0 1 0 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 1 0

28. How many other exams cannot be given on the same day as the exam for course E?

a) 1 b) 2 c) 3 d) 5

29. If a student can take only one exam per day, what is the minimum number of days needed to schedule all of the final exams?

a) 3 b) 4 c) 7 d) 9

30. Two astronauts Pat and Marti were orbitting the earth in separate space capsules, orbitting in the same direction and in the same plane, but at different altitudes. Pat orbits the earth in 3 hours, and Marti in 7.5 hours. At 12 noon Marti sees Pat directly below him. Assuming no factors except their orbital speeds, the next time Marti's capsule is directly over Pat's will be:

a) 4:00 pm b) 5:00 pm c) 7:30 pm d) 8:00 pm