How many sets should be made and sold to maximize the weekly profit?

Question

A manufacturer sells each of his TV sets for 85 dollars. The cost C (in dollars) of manufacturing and selling x TV sets per week is C=1500+10x+0.005x^2. If at most 10,000 sets can be produced per week, how many sets should be made and sold to maximize the weekly profit?

Answer 1

Profit = (Price per set)(number of sets) - Cost for x number of sets)

Profit = 85x - (1500 + 10x + 0.005x²) or P(x) = -1500 + 75x - 0.005x²

To maximize the profit, take the first derivative and set it equal to zero

P'(x) = 75 - 0.01x = 0

x = 7500 TV sets per week is the maximum profit