23. Quadratic Problems

Question

A computer company keeps track of its monthly production and profit over three months. Here is the profit function P(x) = -x^2 +160x - 400, where the profit, P, is a function of x, the number of computers produced in a month.

a. How many computers should be produced to make the maximum profit?

b. What is the maximum profit made in one month?

c. How much profit will be made when 100 computers are produced?

d. How many computers must the company produce to keep from losing money?

Answer 1

x = -b/a/2 at either the maximum or minimum of c a quadratic function: y=ax² + bx + c.

So, x = -160/(-1)/2 = 80.

A. 80 computers.

B. P(80) = -80×80 + 160*80 - 400

-6400 +12800 + 400 = 6400 - 400 = 6000

Maximum profit = $6,000.

C. P(100) = -100×100 + 160×100 - 400 = -10000 +16000 - 400 = $5600

D. There will be a profit when x is at least 3 but no more than 157. I found these bounds by locating the zeros of the profit function.