Question

A gas station sells gallons of gasoline per year, which is delivered times per year in equal shipments of gallons. The cost of each delivery is dollars and the yearly storage costs are T, where T is the length of time (a fraction of a year) between shipments and is a constant.

a. Which value of minimize the costs? Hint: Express T in terms of .

b. Find the optimal number of deliveries if =4 million gal, =$8000, and =31 cents/gal‑yr.

Answer 1

To find the value of N that minimizes the costs, we can take the derivative of the total cost function with respect to N and set it equal to zero. Solving for N gives us N = sqrt(GD/T).

Step-by-step explanation:

To minimize costs, we need to determine the optimal value of T.
Since the gas station sells S gallons per year and receives N equal shipments of G gallons each year, the number of deliveries per year is given by N=S/G.
The storage time T is the length of time (a fraction of a year) between shipments and can be expressed in terms of N as T=1/N. By substituting this expression for T into the equation for the total cost, we can find the value of N that minimizes the costs:

Total cost = (delivery cost per shipment)×(number of deliveries per year) + (storage cost per gallon per year)×(total gallons per shipment)×(storage time per year)

=($D)×(N) + ($T)×(G)×(1/N)

=($D)×(N) + ($T)×(G/N)

Now, to find the optimal value of N, we can take the derivative of the total cost function concerning N and set it equal to zero:
d(Total cost)/dN = $D - ($T)×(G/N^2) = 0

Solving for N gives us:
N = sqrt(GD/T)