Question

A car rental agency rents 190 cars per day at a rate of 29 dollars per day. for each 1 dollar increase in the daily rate, 5 fewer cars are rented. at what rate should the cars be rented to produce the maximum income, and what is the maximum income?

Answer 1

The equation that we can create from this situation is:

i = (190 – 5 x) * (29 + x)

where i is the income and x is the increase in daily rate

Expanding the equation:

i = 5510 + 190x – 145x - 5x^2

i = -5x^2 + 45x + 5510

Taking the 1st derivative:

di/dx = -10x +45

Set to zero to get the maxima:

-10x + 45 = 0

x = 4.5

 

So the cars should be rented at:

29 + x = 33.5 dollars per day

 

The maximum income is:

i = (190 – 5*4.5) * (33.5)

i = 5,611.25 dollars