A box with a rectangular base and open top must have a volume of 128 f t 3 . The length of the base is twice the width of base. The base needs to be stronger than the sides, so it costs more. The base costs $9 per f t 2 , while the sides only cost $6 per f t 2 . We wish to find the dimensions of the box that minimize the cost of material used.

Step-by-step explanation:

Given

Required

The dimension that minimizes the cost

The volume is:

This gives:

Substitute

Make H the subject

The surface area is:

Area = Area of Bottom + Area of Sides

So, we have:

The cost is:

Substitute: and

To minimize the cost, we differentiate

Then set to 0

Rewrite as:

Divide both sides by W

Rewrite as:

Solve for

Take cube roots

Recall that:

Hence, the dimension that minimizes the cost is:

a; x = 0.5

step-by-step explanation:

-10x = -5

x = 0.5