Question

A rectangular box is to have a square base and a volume of 20 ft3. if the material for the base costs $0.28 per square foot, the material for the sides costs $0.10 per square foot, and the material for the top costs $0.22 per square foot, determine the dimensions of the box that can be constructed at minimum cost.

Answer 1

Then the area of the top and bottom are each x^2.
Then the height of the box is 40/x^2.
Then each side of the box is 40/x square feet.
Then the area of all four sides is 160/x square feet.
Then the total cost is represented by:
0.30x^2 + 0.05 (160/x) + 0.20x^2
Expand the middle term and combine x^2 terms:
0.50x^2 + 8/x
Rewrite for easy differentiation:
0.50x^2 + 8x^-1
Take the derivative:
x - 8x^-2
Set the derivative to zero:
x - (8/x^2) = 0
Multiply by x^2
x^3 - 8 = 0
Add 8:
x^3 = 8
Take the cube root:
x = 2 feet
So the height is 40/4 = 10 feet
So the dimensions with minimum cost are:
2 x 2 x 10 feet.