Business Problem: Optimum truck rental price: 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.
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 - Cost = N * (Rent - Cost)
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:
dP
2 * R 55
dR
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