jmc

Question

A shop sells 500 smartphones a week for 450 each. A market survey shows that each decrease of 5 on the price will result in the sale of an additional 10 smartphones per week. What price of the smartphone would result in maximum revenue, in dollars?

Accepted Answer

Suppose the price of the cellphone is reduced to

450 - 5 x

dollars; then it will sell

500 + 10 x

units, so the revenue will be

(450 - 5 x) (500 + 10 x) = 5 (90 - x) 10 (50 + x) = 50 (90 - x) (50 + x) = 50 (- x^{2} + 40 x + 4500),

in dollars.

Completing the square on

- x^{2} + 40 x + 4500,

we get

50 (- x^{2} + 40 x + 4500) = 50 (- (x - 20)^{2} + 400 + 4500) = 50 (- (x - 20)^{2} + 4900) = - 50 (x - 20)^{2} + 245000.

This is maximized when

x = 20,

so the optimal price of the smartphone is

450 - 5 (20) = 350

dollars.