Question

A car rental agency rents 200 cars per day at a rate of ​$30 per day. For each ​$1 increase in​ rate, 5 fewer cars are rented. At what rate should the cars be rented to produce the maximum​ income? What is the maximum​ income?

The rental agency will earn a maximum income of ​$
here when it charges ​$
enter your response here per day.

Answer 1

To find the rate at which the rental agency should rent its cars to produce the maximum income, we can use the following steps:

Let x be the number of $1 increases in rate from the initial rate of $30. Then the new rate will be $30 + x, and the number of cars rented will be 200 - 5x (since for each $1 increase in rate, 5 fewer cars are rented).

The income generated by renting out the cars at this new rate is given by the product of the rate and the number of cars rented, so we can write the income as:

I(x) = (30 + x)(200 - 5x)

Expanding this expression, we get:

I(x) = 6000 + 200x - 150x - 5x^2

I(x) = -5x^2 + 50x + 6000

To find the value of x that maximizes the income, we can take the derivative of I(x) with respect to x and set it equal to zero:

I'(x) = -10x + 50

-10x + 50 = 0

x = 5

Therefore, the rental agency should increase the rate by $5 to maximize its income. The new rate will be $35, and the number of cars rented will be 175. The maximum income can be found by plugging x = 5 into the expression for I(x):

I(5) = -5(5)^2 + 50(5) + 6000

I(5) = $6,250

So the rental agency will earn a maximum income of $6,250 per day when it charges $35 per day.