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:

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: