Maximum Profit? P=24-4x

Question

Amy can produce china pitchers for $6 each. She estimates the price function to be p=24-4x, where p is the price at which exactly x pitchers will be sold per week. Find the number of pitchers that she should produce and the price that she should charge in order to max profit. What is the maximum profit?

Answer 1

Profit = Revenue - Cost

Her revenue is Px=(24-4x)x=24x-4x²

Her cost is 6x

Profit = -4x²+24x-6x= -4x²+18x

Take the first derivative and set it equal to zero, then solve for x

-8x+18=0, 8x=18 x=18/8=9/4 = 2 1/4 pitchers

Profit with 9/4 pitchers = -4(9/4)² + 18(9/4) = -4(81/16) + 162/4 = -81/4+162/4 = 81/4= $20.25 per week

If she produced 2 pitchers, profit would be -4(4)+18(2)=-16+36=$20

If she produced 3 pitchers, profit would be -4(9)+18(3) = -36+54= $18

If fractions of pitchers are allowed her maximum profit is at 2 1/4 pitchers

If no fractions are allowed, she should produce 2 pitchers to maximize profit

Price = 24-4x = 24-4(9/4)= 24-9=$15 price per pitcher

or if x=2, then 24-4(2)=24-8=$16 per pitcher