Bike rentals and maximum income

Question

A rental store charges $32 per day to rent a bicycle. At this rate, the store rents about 200 bikes per month. The owner of the store estimates that they will rent 5 fewer bicycles for each $2 increase in rental price. What price will maximize the income of the store?

Answer 1

Let x = number of $2 increases

 

Let I = income = (number of bikes rented )(price per bike rental)

                          = (200 - 5x)(32 + 2x)

                          = -10x² + 400x - 160x + 6400

                          = -10x² + 240x + 6400

 

The graph of this function is a parabola opening downward.  The maximum occurs at x = -240/(2(-10)) = 12.

 

So, maximum income when the price per rental is 32 + 2(12) = $56.