# 1. A snack food manufacturer

PROBLEM

1. A snack food manufacturer buys corn for tortilla chips from two cooperatives, one in Iowa and one in Illinois. The price per unit of the Iowa corn is \$5.50 and the price per unit of the Illinois corn is \$6.00.

 a. Define variables that would tell how many units to purchase from each source. b. Develop an objective function that would minimize the total cost. c. The manufacturer needs at least 12000 units of corn. The Iowa cooperative can supply up to 8000 units, and the Illinois cooperative must supply at least 6000 units. Develop constraints for these conditions.

2. The relationship d = 5000- 25p describes what happens to demand (d) as price (p) varies. Here, price can vary between \$10 and \$50.

 a. How many units can be sold at the \$10 price? How many can be sold at the \$50 price? b. Model the expression for total revenue. c. Consider prices of \$20, \$30, and \$40. Which price alternative will maximize total revenue? What are the values for demand and revenue at this price?

3. There is a fixed cost of \$50,000 to start a production process. Once the process has begun, the variable cost per unit is \$25. The revenue per unit is projected to be \$45.

 a. Write an expression for total cost. b. Write an expression for total revenue. c. Write an expression for total profit. d. Find the break-even point.

4. An author has received an advance against royalties of \$10,000. The royalty rate is \$1.00 for every book sold in the United States, and \$1.35 for every book sold outside the United States. Define variables for this problem and write an expression that could be used to calculate the number of books to be sold to cover the advance.

5. A university schedules summer school courses based on anticipated enrollment. The cost for faculty compensation, laboratories, student services, and allocated overhead for a computer class is \$8500. If students pay \$420 to enroll in the course, how large would enrollment have to be for the university to break even?

6. As part of their application for a loan to buy Lakeside Farm, a property they hope to develop as a bed-and-breakfast operation, the prospective owners have projected:

 Monthly fixed cost (loan payment, taxes, insurance, maintenance) \$6000 Variable cost per occupied room per night \$ 20 Revenue per occupied room per night \$ 75
 a. Write the expression for total cost per month. Assume 30 days per month. b. Write the expression for total revenue per month. c. If there are 12 guest rooms available, can they break even? What percentage of rooms would need to be occupied, on average, to break even?

7. Organizers of an Internet training session will charge participants \$150 to attend. It costs \$3000 to reserve the room, hire the instructor, bring in the equipment, and advertise. Assume it costs \$25 per student for the organizers to provide the course materials.

 a. How many students would have to attend for the company to break even? b. If the trainers think, realistically, that 20 people will attend, then what price should be charged per person for the organization to break even?

8. In this portion of an Excel spreadsheet, the user has given values for selling price, the costs, and a sample volume. Give the cell formula for

 a. cell E12, break-even volume. b. cell E16, total revenue. c. cell E17, total cost. d. cell E19, profit/loss.
 A B C D E 1 2 3 4 Break-even calculation 5 6 Selling price per unit 10 7 8 Costs 9 Fix cost 8400 10 Variable cost per unit 4.5 11 12 Break-even volume 13 14 Sample calculation 15 Volume 2000 16 Total revenue 17 Total cost 18 19 Profit loss

9. A furniture store has set aside 800 square feet to display its sofas and chairs. Each sofa utilizes 50 sq. ft. and each chair utilizes 30 sq. ft. At least five sofas and at least five chairs are to be displayed.

 a. Write a mathematical model representing the store’s constraints. b. Suppose the profit on sofas is \$200 and on chairs is \$100. On a given day, the probability that a displayed sofa will be sold is .03 and that a displayed chair will be sold is .05. Mathematically model each of the following objectives: 1. Maximize the total pieces of furniture displayed. 2. Maximize the total expected number of daily sales. 3. Maximize the total expected daily profit.

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