1. For most products, higher prices result in a decreased demand, whereas lower prices result in an increased demand. Let

d = annual demand for a product in units
p = price per unit

Assume that firm accepts the following price-demand relationship as being realistic:
d = 800 – 10p
where p must be between $ 20 and $70.
a. How many units can the firm sell at the $ 20 per-unit price? At the $ 70 per-unit price? b. Show the mathematical model for the total revenue (TR), which is the annual demand multiplied by the unit price. c. Based on other considerations, the firm’s management will only consider price alternative that will maximize the total revenue. d. What are the expected annual demand and the total revenue according to your recommended price? Information to solution:

d=800-10pTR = (800 -10p)p

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2. The O’Neill Shoe Manufacturing Company will produce a special –style shoe if the order size is large enough to provide a reasonable profit. For each special-style order, the company incurs a fixed cost of $1000 for the production setup. The variable cost is $ 30 per pair, and each pair sells for $ 40. a. Let x indicate the number of pairs of shoes produced. Develop a mathematical model for the total cost of producing x pairs of shoes. b. Let P indicate the total profit. Develop a mathematical model for the total profit realized from an order for x pairs of shoes. c. What is the breakeven point?

Information to solution:

FC = $1000VC per pair = $ 30

C = FC + VC = 1000 + 30x

R= 40x

П = TR –TC = 40x – (1000 + 30x) = 10x – 1000

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