Acar 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?

The cars should be rented at $37 per day

The maximum income is $ 6845

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:

4. F (x) being a quadratic function, in the form , with , the vertex 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:

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:

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