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A truck rental company rents its 30 trucks by the day. When the rent is $20/day, all 30 trucks are rented. For each $1/day increase in the rental price, they rent one less truck. The cost to the company is $5/day for each rented truck. Find the rental charge which produces the maximum profit.We need to express the profit in terms of the rental charge, cost, and the number rented, N:
Profit = Revenue - Total Cost = N * Rent – N * 5
From the information provided, we can determine that we need to subtract from the 30 available trucks a number equal to the Rent - 20, which represents the non-rented trucks:
N = 30 – (Rent – 20) = 50 – Rent
Now we combine these results to get
Profit = (50 – Rent) * (Rent – 5) = –Rent2 + 55*Rent – 250
Now we have the Profit expressed in terms of 1 variable. We can differentiate to find a maximum:
Setting this to 0 and solving, we find that the optimum rent is $27.50. As a check, the following table confirms this for rounded values:
|
Number of trucks rented |
Rent |
Profit |
|
30 |
$20 |
$450 |
|
24 |
$26 |
$504 |
|
23 |
$27 |
$506 |
|
22 |
$28 |
$506 |
|
21 |
$29 |
$504 |