math exam

Name__________________

1.

a. (5 pts) Which of the following mappings are functions, and of those that are functions,

which are invertible?

b. (10 pts) Identify the domain, horizontal, and vertical asymptotes of the following function:

𝑓(𝑥) = 𝑥2 − 3𝑥 + 2

2𝑥2 − 2

𝑥 1 2 3 4

𝑓(𝑥) 8 5 6 7

𝑥 1 2 2 4

𝑓(𝑥) 5 6 7 8

𝑥 1 2 3 4

𝑓(𝑥) 7 7 6 5

𝑥 1 2 3 4

𝑓(𝑥) 3 2 4 1

2. (15 pts) A jeweler makes a new line of rings, bracelets, and pendants that combine silver, gold,

and platinum. The rings each require 4 grams of silver, and 3 each of gold and platinum. The

bracelets require 15 grams of silver, 10 of gold, and 5 of platinum. The pendants each require 10

grams of silver, 20 of gold, and 10 of platinum. If the jeweler uses 223 grams of silver, 286 grams

of gold, and 161 grams of platinum, how many of each did he make?

3. (20 pts) We model the economy of a municipality by dividing it into 2 categories: goods and

services. It requires $0.20 of goods and $0.25 of services to create $1.00 of goods. It requires

$0.20 of goods and $0.10 units of services to create $1.00 of services. How much money would

they have to spend on goods and services to satisfy an external demand for $12,000 worth of

goods and $8,000 worth of services?

4. (20 pts) An investor wishes to buy stock in two companies: Acme and Bantam. Acme’s shares

are priced at $50 and Bantams are priced at $20. He believes, in the worst-case scenario, that

Acme’s shares could drop by as much as $2 and Bantam’s by $8. He believes that, in the most

likely scenario, Acme’s shares will go up by $3 and Bantam’s by $6. He has $60,000 to invest,

that he doesn’t wish to lose any more than $10,000 in the worst case scenario, and that he

doesn’t wish to buy more than 1,400 shares of both stocks combined. He wishes to maximize his

earnings in his most likely scenario.

a. (15 pts) What are the corner points in this problem?

b. (5 pts) What should his strategy be in order to maximize his earnings in the most likely

scenario?

5. (15 pts) A 40-year-old man wishes to start saving for his retirement. At the end of each month,

he starts depositing 𝑅 dollars in an account bearing 5% interest compounded monthly. After 15

years, he doubles the amount he deposits each month. Once he turns 70, he stops making

deposits and uses all the money in the account to purchase a 20-year ordinary annuity which

make monthly payments of $20,000, valued at 6% interest compounded monthly. How much

should 𝑅 be to ensure he can afford this annuity?

6. (15 pts) A farmer is raising free-range pigs, cows, and sheep. Pigs require a monthly budget of

$25 for food, grooming, and health upkeep, 3 acres of land, and 30 gallons of water weekly.

Cows have an upkeep budget of $40, require 5 acres, and 20 gallons of water weekly. Sheep

require a budget of $50, 2 acres, and 25 gallons of water. The farmer can sell the pigs for a profit

0f $215 each, the cows for $350, and the sheep for $250, and wishes to maximize his profit. He

considers this a linear programming problem and uses Excel to create the following sensitivity

report:

a. (4 pts) How many of each type of livestock should he raise in order to maximize his profits?

b. (4 pts) Suppose his profit per sheep were to change dramatically. How much could it go up or

down before the farmer should adopt a new strategy?

c. (3 pts) How much more profit could the farmer make per acre if he had more acres of land?

d. (4 pts) How many more acres could he accommodate before this answer changed?

Variable Cells

Final Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease

$B$2 pigs 37.14 0 215 111.47 1.76

$C$2 cows 4.62 0 350 3.00 150.00

$D$2 sheep 7.73 0 250 60.00 105.28

Constraints

Final Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease

$F$4 feed/care budget 1500 3.18 1500.00 1300.00 255.56

$F$5 acreage 150 44.12 150.00 32.86 15.71

$F$6 water allotment 1400 0.10 1400.00 137.50 650.00