Calculus Problem Optimization

Question

A box with a square base and open top is to have a volume of 128000 cubic inches. The material for the base costs 4 cents per square inch, while the material for the sides costs 8 cents per square inch. Find the dimensions which minimize the cost of material.

Height =

Edge of Base =

Answer 1

The first thing we do is create our volume function.

V = B^2 * H

where B is the edge length of the base, and, H is the height

V = 128000

Therefore,

B^2 * H = 128000

Now we will want to rewrite H in terms of B.

H = 128000/ B^2

The next step is to create our cost function.

C = cost of base + cost of the sides

C = cost of base + 4 * cost of one side

C = cost per square inch of base * area of the base + 4* cost per square inch of side * area of one side

C = 4*B^2 + 4*8*B*H

Now we will substitute our H(B) function into our cost function.

C(B) = 4*B^2 + 4*8*B*(128000/ B^2)

Simplifying,

C(B) = 4B^2 + 32*128000/B

Now we take the derivative of C with respect to B.

C' = (8 (-512000 + B^3))/B^2

Now we set C' = 0 because we want to find the maximum of C (the point on C where there is no slope)

0 = (8 (-512000 + B^3))/B^2

B = 80 cm is the only real solution.

Now we solve for H

H = 128000/(80^2)