The profit of a shoe company is modelled by the function P(x) = -5(x – 4)2 + 45, where x is the number of pairs of shoes produced in thousands, and P(x) is the profit, in thousands of dollars.

Question

The profit of a shoe company is modelled by the function P(x) = -5(x – 4)2 + 45, where x is the number of pairs of shoes produced in thousands, and P(x) is the profit, in thousands of dollars. How many thousands of pairs of shoes will the company need to sell to earn a profit?

Answer 1

One way to solve this problem is to determine the breakeven point. “Breakeven” means that the seller neither loses or makes money; his cost is exactly equal to his revenue.

This problem tells us that P(x) is profit. If we set P(x) =0, we can solve for the number of pairs of shoes in thousands that must be sold to break even.

P(x) = -5(x-4)2 + 45

0 = -5(x-4)2 + 45

0 = -10(x-4) + 45

0 = -10x + 40 + 45

0 = -10x + 85

10x = 85

x = 8.5

Breakeven is 8500 pairs of shoes. Therefore, the company must sell at least 8,501 pairs of shoes to make a profit.