#### DEPARTMENT OF MATHEMATICS & STATISTICS MATH 1833

FINAL EXAMINATION
DECEMBER 1996 TIME: 3 Hours

#### Instructions:

(b) Show all intermediate calculations.

#### FORMULAS

1. It costs a trailer manufacturer \$14,400 to make 20 trailers, but only \$8,000 to produce 10 trailers. The trailers are sold at \$800 each. Assume the cost, revenue and profit functions are linear.

 (a) Find the cost function. (b) Find the revenue and profit functions. (c) How many trailers must be made and sold to break even?

2. If the supply and demand are given by:

 supply: 2q - 6p = 48 demand: 3q + 5p = 660

find the price and quantity at market equilibrium.

3. Set up the following as a system of linear equations to be solved. Do not solve the system.

A company produces 2 models of bicycles, model 201 and 301. Model 201 requires 2 hours of assembly time and model 301 requires 3 hours of assembly time. The parts for model 201 cost \$25 per bike and the parts for model 301 cost \$30 per bike. If the company has a total of 34 hours of assembly time and \$365 available per day, how many of each can be made in a day?

4. Solve the following system of equations using matrices and Gauss-Jordan elimination. Show all intermediate steps.

5. For the matrices and , find the following where possible:

(a) A + B (b) AB (c)BA (d)

6. Find the inverse of the following matrix:

7. Solve the following linear programming problem:

A part-time farmer wants to raise geese and pigs. He wants to raise no more than 16 animals, including no more than 10 geese. It costs \$5 to raise a goose and \$15 to raise a pig and he has \$180 for this project. Each goose produces \$6 in profit and each pig \$20 in profit. How many of each animal should be raised to maximize the profit?

8. In a genetics experiment on pea plants, it is found that:

 22 are tall; 25 have white flowers; 39 have smooth skins; 9 are tall and have white flowers; 17 are tall and have smooth skins; 20 have white flowers and smooth skins; 6 have all 3 characteristics; 4 have none of these 3 characteristics.

 (a) How many plants are there? (b) How many are tall but have neither white flowers nor smooth skins? (c) How many are not tall but have white flowers and smooth skins?

9. In Canada, radio station call letters consist of 4 letters with the first letter always a C. How many sets of call letters are possible if

 (a) no letter may be repeated? (b) repeats are allowed? (c) consecutive letters must be different?

10. In Lotto 6-49, a person selects 6 different numbers of the numbers from 1 to 49. How many tickets with exactly 2 of the 6 winning numbers can be formed?

11. In a shipment of calculators, 8% have defective cases, 11% have defective batteries and 3% have both defects. What is the probability that a calculator chosen at random has neither defect?

12. If 3 cards are drawn at random without replacement from a standard desk of 52 cards, what is the probability that at least 1 of the cards is red?

13. A study found that 49% of those involved in a fatal car accident wore seat belts. Of those in a fatal crash who wore seat belts, 44% were injured and 27% were killed. Of those in a fatal crash who did not wear seat belts, 41% were injured and 50% were killed.
 (a) What is the probability that a randomly selected person that was killed in a car crash was wearing a seat belt? (b) What is the probability that a randomly selected person who was unharmed in a fatal crash was not wearing a seat belt?

14. If tuition of \$3538 is due in 4 months, how much should be invested in an account paying 6.25% simple interest?

15. What lump sum should be invested in a savings account to yield \$20,000 in 5 years at 8% interest compounded quarterly?

16. How much money will accumulate in a retirement account over 9 years if \$1,000 is deposited semi-annually at 8% compounded semi-annually?

17. A house costing \$285,000 was purchased with a \$60,000 down payment.
 (a) What are the monthly mortgage payments on the remainder if the mortgage is a 20-year mortgage at 9% compounded monthly? (b) How much of the mortgage is left after 5 years have gone by?