Consider a pizzeria that sell pizzas for a revenue of R(X)=ax and costs C(X)=b+cx+dx^2,where x represents the number of pizzas.

Question

Assume that R(x) = 16x and C(x) = 2x + x². How many pizzas sold maximizes the profit?

Answer 1

P(x) = R(x) - C(x)

Determine x for

P'(x) = 0

Answer 2

Consider a pizzeria that sell pizzas for a revenue of R(X)=ax and costs C(X)=b+cx+dx^2,where x represents the number of pizzas. Assume that R(x) = 16x and C(x) = 2x + x². How many pizzas sold maximizes the profit?

Detailed Solution:

Profit = Revenue – Cost ==> P(x) = R(x) – C(x) ==> P(x) = (16x) – (2x + x²) ==> P(x) = 16x – 2x – x² ==> P(x) = 14x – x²

Take Derivative of Profit and set it equal to zero; then solve for x which is number of Pizzas sold that maximizes the profit:

P'(x) = 14 – 2x = 0 ==> 2x = 14 ==> x = 7 (number of Pizzas sold that maximizes the profit)

Maximum Profit: P(x) = 14x – x² ==> P(7) = 14(7) – (7)² = 7(14 – 7) = (7)(7) = $49 (Maximum Profit)