Name:________________________

Final Learning Motivator- Quantitative Economics 39

May 26, 1997

Please show all of your work clearly in the space provided. Illegible writing will be marked incorrect. If you use the back of a page, indicate that you have done so on the front of the page. Good Luck!

For numerical answers, please circle the number you would like to have graded.

1) If there are four airlines that travel from New York to Washington and two that travel from Washington to Richmond, how many ways could a flight be booked from New York to Richmond through Washington? Draw the associated tree diagram. (11-1 #12)

 

 

 

 

 

 

 

 

2) A testing service reports the average score on a certain test to be 200 points. s = 25 points. (12-9 #5, b,c,d,e).

b) What percentage of those taking the test score below 160?

 

 

 

 

c) Half of those taking the test make scores in what interval symmetrically located above and below 200?

 

 

 

 

d) Ninety-five percent of those taking the test score above what number of points?

 

 

 

 

  1. What would be the percentile standing of a person who scored 260 points on the test?

 

3) Assume a store’s profit is dependent on the number of salespersons, s, and the amount of inventory, i (in hundreds of dollars). If the profit is given by

P(s, i) = 1,400 - (12 - s)2 - (40 - i)2,

then what values of s and i will maximize profit? Find the maximum profit. (9-5 #15)

 

 

 

 

 

 

 

 

4) (8-1 #28) The profit realized when y gallons of distilled water are made and sold is

P(y) = 20y - 0.005y2.

  1. Find the number of gallons that should be made to maximize profit.

 

 

 

 

 

 

 

 

 

 

 

 

 

  1. Prove that (a) is a maximum.

 

 

 

 

 

 

 

 

 

 

5) A consulting firm conducts training sessions for employees of various companies. The charge to a company sending employees to a session is $50 per employee, less $0.50 for each employee in excess of 10. That is, for example, if 12 employees are sent, the charge per employee would be $49.00 and the total prorated charge to the company would be 12(49.00) = $588.00. The consulting firm further has a fixed total charge for groups of x or more, where x is the number that maximizes the prorated group charge. What should x be, and what is the maximum total group charge to a company? (8-3 #20).

 

 

 

 

 

 

 

 

 

 

6) a) Convert f(x) = 2 - x--2, , to a probability density function. (12-5 #4)

 

 

 

 

 

 

 

 

b) Compute the probability that a randomly selected x will lie in the interval .

 

 

 

 

 

 

 

7) Compute and s for the following sample of x’s: 8, 8, 9, 6, 4. Please write out the equations you use to find the answer. (12-8 #1).

 

 

 

 

 

 

 

8) A local weekly newspaper receives orders for one-, two-, and three-year subscriptions, with probabilities ¼, ½, and ¼ respectively. For each subscription year, it receives $2. What is the expected return for each order received? (12-2 #10)

 

 

 

 

 

9) Suppose the probability that a person has a disease D is 0.09, the probability that a medical examination will indicate the disease if a person has it is 0.6, and the probability that medical examination will indicate the disease if a person does not have it is 0.05. What is the probability that a person has the disease if medical examination so indicates? (11-5 p.800-1)

 

 

 

 

 

 

 

 

 

 

 

10) An urn contains 600 glass balls. Each ball has a left half and a right half of different colors. The left half may be G or Y (for green or yellow) and the right half may be R, W, or B (for red, white, or blue). The number of balls of various colors are shown in the table. (ll-3 #4)

 

R

W

B

G

30

42

138

Y

270

78

42

a) Are white and blue mutually exclusive? Why?

 

 

b) Are white and blue independent? Why?

 

 

  1. Are yellow and white mutually exclusive?

 

 

  1. Are yellow and white independent?

11) A company operates a fleet of delivery trucks. Study shows that gallons of fuel consumed per mile of driving, F(x), is related to the speed at which a truck is driven, x miles per hour, by the function

,

where k1 and k2 are parameters (constants) that vary somewhat from truck to truck. (8-4 #11)

  1. Find and write the expression for the speed, x, that will lead to minimal fuel consumption per mile of driving.

 

 

 

 

 

 

 

b) What speed will provide minimal fuel consumption per mile of driving for a truck having k1=4.9 and k2=0.004?

 

 

12) (11-1 #31) a. How many different ways can a committee of 7 be chosen from a school senate of 63?

 

 

 

 

 

(#15) b. If a true-false examination has 20 questions, how many different answer sheets could there be?

 

 

 

 

 

13) (11-2 #15) A pizza shop offers cheese pizza with any combination of six other toppings: hamburger, onions, mushrooms, pepperoni, peppers, and sausage. How many different kinds of pizza can be ordered?

 

 

 

 

 

 

 

 

14) (10-4 #17) The supply and demand functions for a product are

ps(q) = 10 + 0.1q and

pd(q) = 100 - 0.2q,

where q is in thousands of tons and price is in dollars per ton.

    1. Compute consumers’ surplus.

 

 

 

 

 

 

 

 

b) Compute producers’ surplus.

 

 

 

 

 

 

 

 

15) Find the inverse if it exists: (2-5 #1)

 

 

 

 

 

16) Draw the Venn diagram and use the given information to determine the number of elements in each region: (11-1 #3)

 

 

 

 

 

 

 

 

17)

  1. What sum of money deposited now at 8 percent compounded quarterly will provide just enough money to pay a $1000 debt due 7 years from now? (6-3 #5)

 

 

 

 

 

 

  1. New Venture Corporation has decided to transfer a sum of money to a reserve account at the end of each year to accumulate $100,000 to be used to replace machinery10 years from now. [Perhaps your company will want to do this to replace computers.] How much should be transferred each year if interest at 8 percent compounded annually is credited to the reserve? (6-6 #9)

 

 

 

 

 

 

 

18) Stay Trim Bakery makes two kinds of diet candy: Luscious and Delicious. To make one dozen of Luscious Candy requires 1 gallon of milk, 2 pounds of butter, and 1 pint of cream. To make one dozen of Delicious Candy requires 3 gallons of milk, 1 pound of butter, and 1 pint of cream. Find the number of dozens of each candy to be made in the coming week if exactly 24 gallons of milk, exactly 13 pounds of butter, and exactly 10 pints of cream are to be consumed. (2-7 #17)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

19) (1-5 #14) A manufacturer has a fixed cost of $120,000 and a variable cost of $20 per unit made and sold. Selling price is $50 per unit.

  1. Find the revenue, cost, and profit functions using q for the number of units.

 

 

 

 

 

b) Compute the profits if 10,000 units are sold.

 

 

  1. Find the break-even quantity.

 

 

e) Find the break-even dollar volume of sales (revenue).

 

 

f) Construct the break-even chart. Label the cost and revenue lines, the fixed cost line, and the break-even point.

 

 

 

 

 

 

 

 

 

 

 

20) Find the limit if it exists: (7-2 #26)

.