A local beer distributor contracts with a nearby brewery to produce its signature brand of beer.

Question

A local beer distributor contracts with a nearby brewery to produce its signature brand of beer.

The distributor needs an amount of 8000 barrels of beer to meet their annual demand. Since the demand is uniform throughout the year, the distributor can reduce storage costs by dividing that amount into multiple smaller runs.

The annual cost of storage can be computed with the function ƒ where x is the number of runs per year.

ƒ(x) = 50(8000/x) dollars

The brewery collects a flat surcharge of $1000 every time the brewery produces a run of the signature brand.

The annual cost of running the brewery can be computed using the function g where x is the number of runs per year.

g(x) = 1000x dollars

Write a function C(x) for the combined cost of storage and running the brewery.

C(x) = ? dollars

What is the number of runs x that will minimize the combined cost?

x = ?

Answer 1

Hi,

c(x) = total costs that includes cost of storage and new brand = 1000x + 50(8000/x)

c'(x) = 1000 +50( -8000/x^2)

c'(x) = 0 ====> 1000 -50( 8000/x^2) = 0 =====> 1000 = 50( 8000/x^2)=====> x^2 = 400 ====> x = 20

I hope it is useful,

Minoo