Question 1056801
Profit is the difference between revenue and cost.


With a price of p = 140 - 0.01x, the sale of x headphones gives revenue of:


R(x) = px = (140 - 0.01x)x = 140x - 0.01x^2


The costs are:


C(x) = 40x + 15000


The profit is then:


P(x) = R(x) - C(x) = (140x - 0.01x^2) - (40x + 15000)


P(x) = -0.01x^2 + 140x - 40x - 15000


P(x) = -0.01x^2 + 100x - 15000


Taking the derivative of P(x) gives:


P'(x) = -0.01(2)x + 100


P'(x) = -0.02x + 100


Setting the derivative equal to 0 and solving for x then gives:


P'(x) = -0.02x + 100 = 0


0.02x = 100


x = 100 / 0.02


x = 5000


The maximum profit corresponds to the production of 5000 headphones.