Calculus I Optimization

Question

A rectangular storage container with an open top is to have a volume of a  cubic meters. The length of its base is twice the width. Material for the base costs a  dollars per square meter. Material for the sides costs c  dollars per square meter. Find the cost of materials for the cheapest such container.

Minimum cost = ?

Answer 1

Let the width be x so that the length is 2x and height is a/(2x²).

 

The cost for the container is then:

 

C(x) = a·x·2x + c·[a/(2x)]·2(x + 2x) = 2ax² + 3ac/x

 

To minimize the cost, set C'(t) = 0.

 

C'(t) = 4ax - 3acx^-2

 

4ax = 3acx^-2

 

x³ = 3c/4

 

x = ³√(3c/4)