A rectangular box is to have a square base and a volume of 40 ft3. If the material for the base costs $0.34 per square foot, the material for the sides costs $0.05 per square foot, and the material for the top costs $0.16 per square foot, determine the dimensions of the box that can be constructed at minimum cost.

The dimensions of the box so that total costs are minimum are a side length of 2 feet and a height of 5 feet.

Step-by-step explanation:

Geometrically speaking, the volume of the rectangular box (), in cubic feet, is represented by this formula:

(1)

Where:

- Side length of the box, in feet.

- Height of the box, in feet.

In addition, the total cost of the box (), in monetary units, is defined by this formula:

(2)

Where:

- Unit cost of the base of the box, in monetary units per square foot.

- Unit cost of the top of the box, in monetary units per square foot.

- Unit cost of the side of the box, in monetary units per square foot.

By (1), we clear into the expression:

And we expand (2) and simplify the resulting expression:

(3)

If we know that , , and , then we have the resulting expression and find the critical values associated with the side length of the base:

The first and second derivatives of this expression are, respectively:

(4)

(5)

After equalizing (4) to zero, we solve for : (First Derivative Test)

Then, we evaluate (5) at the value calculated above: (Second Derivative Test)

Which means that critical value is associated with minimum possible total costs. By (1) we have the height of the box:

The dimensions of the box so that total costs are minimum are a side length of 2 feet and a height of 5 feet.

7

step-by-step explanation:

the answer would be 7