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

Answer:

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:

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

$F(x)=6600-150x+220x-5x^{2}$

$F(x)=-5x^{2} +70x+6600$

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:

$h=\frac{-b}{2a}$

$h=\frac{-70}{2(-5)}$

$h=7$

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