Question

A car rental agency rents 220 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?

Answer 1

The cars should be rented at $37 per day

The maximum income is $ 6845

Step-by-step explanation:

To solve this problem, a model like this is proposed:

1. If for $ 1 increase in rate 5 fewers cars are rented, then for X monetary units 5X fewe cars are rented.

2. This expressed in rate and cars terms is:

Rate= 30 + x

Cars= ( 220 - 5x)

3. And in the income formula is:

Income=Rate X Cars

But the income formula in function of x monetary units [F(x)] is:

`F(x)=(220-5x)(30+x)`

`F(x)=6600-150x+220x-5x^(2)`

`F(x)=-5x^(2) +70x+6600`

4. F (x) being a quadratic function, in the form
`f(x)=ax^(2) +bx+c`, with
`a<0` , the vertex
`(h,k)`is determined because is the point of maximum values of the function.

The value of x of the vertex will give us in how many monetary units can be increased the rent.

5. We find the value of h with the following formula:

`h=(-b)/(2a)`

`h=(-70)/(2(-5))`

`h=7`

6. We replace the value of h in rate formula:

Rate= 30+h

Rate= 30 + 7

Rate= 37

And we respond the first question: The cars should be rented at $37 per day

7. We replace the value of h in F(x) or Income formula:

`F(x)=-5x^(2) +70x+6600`

`F(x)=-5(7)^(2) +70(7)+6600`

`F(x)=-245 +490+6600`

`F(x)= 6845`

or

Income= Rate X Cars

Income = 37 X (220 - 5(37))

Income= 37 X 185

Income= 6845

And we respond the second question: The maximum income is $ 6845