profit function

Question

For a certain company, the cost function for producing x items is C(x)=40x+200 and the revenue function for selling x items is R(x)=−0.5(x−90)2+4,050. The maximum capacity of the company is 120 items.

Answer 1

Revenue = R = -(1/2)(x^2 -180x +8100) + 4050 = -x^2/2 +90x -4050+4050 = -x^2 +90x

Profit = R-C = -x^2 +90x -40x -200 = -x^2 +50x -200

P' = R'-C'= -2x +50 = 0

2x =50

x =25 as the profit maximizing level of production, which is well below max capacity

maximum profit = P(25) = -625+1250-200 = $450