𝑝(𝑥) = −2𝑥^2 + 200𝑥 + 25,000

Question

A company that produces cellular phones analyzes its production and finds that the profit P (in dollars) on the sale of the phones is approximated by the function. where x is the number of phones produced each month. How many phones per month should be made so as to maximize profit? And what is the maximum profit? Please explain each step thank you!

Answer 1

This is a parabola opening down so the vertex will give us the maximum value.

Use x = -b/2a. to get x= -200/(2x-2) = -200/-4. = 50

So 50 phones should be made. To find the profit substitute x=50 into the original equation.

This gives y =. -2 (50)^2 + 200(50). +. 25000. = 30000

Answer 2

0 = dP/dx = -4x+200

-200/-4 = x

x = 50

p(50) = -2(2500) + 200*50 + 25000

= -5000 + 10000 + 25000

= 30k