Find the revenue function, and calculate the optimal price that would maximize the profit.

Question

The marketing department provides the following price-demand equation: p=120-0.05x

The cost of producing x items is C(X)=500+5x

Find the revenue function, and calculate the optimal price that would maximize the profit.

Answer 1

Revenue R(x) = xp = x(120 - 0.05x) = 120x - 0.05x²;

Profit P(x) = R(x) - C(x) = 115x - 500 - 0.05x^2;

P'(x) = 115 - 0.1x = 0; x = 1150

P''(x) = - 0.1 < 0. So, at x = 1150 we have maximum profit and price is p = 120 - 1150·0.05 = 62.50