Maximizing profit

Question

The weekly demand for a certain type of dog treat approximately satisfies the equation p=16-2x and the total cost of producing x units of dog treats is C(x)=50+4x. The quantity x is measured in thousands of pounds, and the price and total cost are measured in thousands of dollars. Determine the level of production and the corresponding price that maximize the profit.

Answer 1

p=16-2x

R=px= 16x-2x^2

C= 50+4x

Profit=P=R-C= 16x-2x^2 -(50+4x)= -2x^2+12x-50

P'= -4x+12=0

x= 3= 3,000 lbs

P(3)= -18+36-50= -$32,000= max profit or min loss

-2x^2+12x-50

=-2(x^2-6x +9)-50+18

= -2(x-3)^2 -32

vertex=max = (3,-32)=(3000, -32,000)

p=16-2x= 16-2(3)=10= $10,000=profit maximizing price