Topical Problems -- WSMC Regional Contest -- 3/18/1998
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 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.
In 1960 the Olympic record in women's 400m freestyle swimming was 290.3 sec (USA). If the record in 1992 was 246.88 sec (GER), and the record had decreased linearly:
1. At what rate (sec/yr) did the record improve?
a) -1.36 b) - 4.54 c) -7.72 d) - 43.42
2. What would the record have been in 1944, if the Olympics had been held?
a) 312 b) 317 c) 321 d) 326
3. If this linear model continues to hold, during what year (Olympic or not) will the record go under 3 minutes?
a) 2039 b) 2040 c) 2041 d) 2042
During one season, a basketball team played 180 games. The longest winning streak was 5 games, and the longest losing streak was 3 games.
4. The number of games they won must have been:
a) at most 36 b) at most 45 c) at most 135 d) at most 148
5. If any winning streak length from 1 to 5 were equally likely, and any losing streak length from 1 to 3 were equally likely, how many season wins would they expect to have?
a) 100 b) 108 c) 113 d) 120
A company wants to create an outdoor display area for its machinery. Three sides of the rectangular area will be made from chainlink fencing costing $9.00 per foot, and the fourth side will be formed by a brick wall costing $18.75 per foot.
6. If the area to be enclosed is 1000 ft2, what is the minimum cost of enclosing the area?
a) $1268.41 b) $1311.09 c) $1378.38 d) $1415.03
7. If the company purchased 1000 ft of fencing on sale to use on three sides, and used the side of an existing building as the fourth side, what would be the maximum area (in ft2) that could be enclosed?
a) 62,500 b) 111,111 c) 125,000 d) 250,000
8. If the company was also able to secure 100 rigid sections of fence, each 10 feet long, and used only these sections to enclose an additional area at another site away from the building, what would the maximim of this area be? (in ft2)
a) 60,000 b) 62,500 c) 79,600 d) 100,000
A candy company has created 3 special candies to use in their Easter mixes: Peanut butter Chocolate in the shape of bunnies, Marshmellow Cherry creams in the shape of eggs, and Chocolate Nut in the shape of chicks. The table below lists the composition of each mix, and the calorie content and cost to buy each type of candy.
| Special Mix | Deluxe Mix | Colossal Mix | Calories per piece | Cost per piece in cents | |
| Pb. C. bunnies | 8 | 15 | 18 | 75 | 15 |
| M. C. eggs | 8 | 8 | 10 | 60 | 8 |
| C. N. chicks | 8 | 12 | 20 | 70 | 12 |
The company receives an order for 350 packages of Special Mix, 200 of Deluxe Mix, and 180 of Colossal Mix.
9. The total calories in the order is closest to:
a) 1,500,000 b) 1,700,000 c) 1,900,000 d) 2,000,000
10. The order is large enough to qualify for a 5% discount. With an 8.6% sales tax (after discount), and $2.50 shipping charge per $100, or part of $100, ordered (added last), what will be the total cost of the order?
a) 2435.84 b) 3077.68 c) 3090.19 d) 5268.44
11. The company advertises a new mix, the Economy mix, that will enable them to use remnants of production runs of the other mixes and will be the average of the three other mixes and priced at one-half the cost of this average mix. In order to meet the 'truth in advertising standards', (ie, that the product must be at least as good as advertised,) what will be the company's pricing for this mix ?
a) $2.25 b) $2.17 c) $2.08 d) $2.01
Given the function f (3n ) = n + f (3n - 3), and that f (3) = 1:
12. If n is any real integer greater than 1, the value of f ( 21) is:
a) 6 b) 7 c) 21 d) 28
13. If the values generated by using the recursive definition above are used to generate a polynomial function model, the value of f (1) is:
a) .111... b) .144 c) .222... d) .333...
A flagpole is centered in a rectangular slab of concrete whose length and width are 5 m. and 8 m. It is tethered to the ground by four wires, each of length 12 m. and attached from the midpoint of the flagpole to a vertex of the rectangle.
14. How tall is the flagpole?
a) 11 m. b) 18 m. c) 22 m. d) 24 m.
15. What angle does a wire make with the flagpole?
a) 20° b) 23° c) 30° d) 45°
A 4-H member raises only geese and pigs. She wants to raise no more than 16 animals, including no more than 10 geese. She spends $5 to raise a goose and $15 to raise a pig, and she has $180 available for this project. The 4-H member wants to maximize her profit.
16. If each goose produces $6 in profit and each pig $20 in profit, how many pigs should she raise?
a) 0 b) 6 c) 10 d) 12
17. If the profit on a goose doubles and on a pig drops by half, how many pigs should she raise?
a) 0 b) 6 c) 10 d) 12
Five forest rangers are working in an area where communication is difficult. The matrix below describes the one way single communication paths that are possible. A ranger can also act as a relay between two other rangers. In the matrix the communication path goes from the row to the column, where a 1 signifies that communication is open and a 0 that it is closed.
| A | B | C | D | E | ||
F R O M |
A | 1 | 1 | 0 | 0 | 1 |
| B | 0 | 1 | 1 | 0 | 1 | |
| C | 0 | 1 | 1 | 1 | 0 | |
| D | 0 | 0 | 1 | 1 | 0 | |
| E | 0 | 1 | 0 | 0 | 1 |
18. How many one-way ranger-to-ranger communication paths (of any length) do not exist?
a) 4 b) 6 c) 8 d) 12
19. Which matrix element changed from a 0 to a 1 would enable all rangers to communicate with each other? [ pairs are (row# , column#) ie. A to A is (1,1) ]
a) (2,1) b) (4,5) c) (5,3) d) (5,4)
In her latest game, Gina bowled a 199 game, and raised her season average from exactly 177 to exactly 178.
20. In order to raise her season average to exactly 179 with the next game she bowls, what must her next game score be?
a) 180 b) 199 c) 200 d) 201
21. If in her next 5 games (including the game above) she averages exactly 189, her new season average is:
a) 180 b) 182 c) 184 d) 186
22. If the averages 177 and 178 were not exact, but rounded to the nearest integer, what would have been the minimum number of games she could have played to have her average change from 177 to 178 by bowling a game of 199?
a) 9 b) 10 c) 11 d) 12
A space station has the shape of a cube, each of whose edges is of length s. An astronaut working on the outer surface is anchored by means of a rope which allows him to reach a distance s from the anchor point.
23. The surface area of the station accessible to the astronaut, in terms of s , if the anchor point is located at one corner of the cube is:
a) 32 s2 b) 3s2 c) 2s2 d) 34 s2
24. The surface area of the station accessible to the astronaut, in terms of s , if the anchor point is located in the center of a face of the cube is approximately:
a) 3s2 b) 2.8s2 c) 2.5s2 d) 2s2
You are the manager of a scholarship fund for prospective math teachers. At the beginning of each year, $5000 is awarded in scholarships. When you begin your term as manager at the beginning of the year, the scholarships for that year have already been awarded and the fund has assets of $100,000, which you invest at 6% per year compounded monthly .
25. How much does the fund contain at the end of your fifth year as manager? (ie., you have given out scholarships 5 times and are starting your sixth year)
a) $105,640 b) $105,840 c) $106,600 d) $109,890
26. As you assume the manager position, the Board of Directors suspends the scholarship awards temporarily and directs you to invest $70,000 conservatively at 6% and $30,000 more agressively at 17%, both compounded continuously. The scholarships will be reinstated when the two funds are equal in value. How many years will you have to wait to award your first batch of scholarships?
a) 6 b) 7 c) 8 d) 9
A circular dartboard has a circular center of radius 1 in. and adjacent, concentric rings of width 1 in. There is no area on the dartboard outside of the outer ring. The point values for the center and rings from inner to outer are 5, 4, 3, 2, and 0. Assume that when a dart is thrown, it hits the dartboard randomly and sticks (scores).
27. In how many ways can a score of 30 be made by throwing 7 darts?
a) 3 b) 4 c) 5 d) 6
28. If two darts are thrown, what is the probability that the score is 7?
a) .80 b) .64 c) .33 d) .07
The interior angles of a polygon form an arithmetic series whose first term is 120 and third term is 130.
29. How many sides does the polygon have?
a) 9 b) 10 c) 12 d) either 9 or 16
30. What angle 0° < Q < 180° would have the same cosine value as the cosine value of the 68th term in the sequence?
a) 95° b) 115° c) 125° d) 145°