A rectangular storage container with an open top is to have a volume of 10 m^3. The length of its base is twice the width. Material for the base costs $6 per m^2. Material for the sides costs $0.8 per m^2. Required:
Find the dimensions of the container which will minimize cost and the minimum cost.

The dimension that minimizes the container is width = 1; base = 2 and height = 5

The minimum cost is $36

Explanation:

Let the width be x

So:

Volume of the box is:

---- Given

Volume is calculated as:

Substitute 10 for Volume

Make h the subject

Next, we calculate area of the sides.

Because it has an open top, the area is:

The base area costs $6 per m²

So, the cost of 2x² would be:

The side cost area costs $0.8 per m²

So, 6xh would cost

Total Cost (C) is:

Recall that

So:

Take derivative of C

Take LCM

Equate to 0

Cross multiply

Add 24 to both sides

Divide through by 24

Take cube roots of both sides

Recall that

and

Solve for these dimensions:

i.e.

Hence, the dimension that minimizes the container is width = 1; base = 2 and height = 5

Recall that

Substitute 1 for x

The minimum cost is $36