A rectangular box is to have a square base and a volume of 32 . If the material for the base costs 10 cent/square foot, the material for the sides costs 30 cent/square foot, and the material for the top costs 20 cent/square foot, determine the dimensions of the box that can be constructed at minimum cost.

Question

A rectangular box is to have a square base and a volume of 32  . If the material for the base costs 10 cent/square foot, the material for the sides costs 30 cent/square foot, and the material for the top costs 20 cent/square foot, determine the dimensions of the box that can be constructed at minimum cost.

Quizplus · Accepted Answer

Let the length and width of the square base be x. Then the height of the rectangular box is 32/x^2.
The cost of the base is 10x^2 cents.
The cost of the top is also 10x^2 cents.
The cost of each side is 2x times the height, which is 64/x.
So the cost of the sides is 128x cents.
The total cost is therefore C(x) = 20x^2 + 128x.
To minimize this function, we take its derivative and set it equal to zero:
C'(x) = 40x + 128 = 0
x = 64/5
Substituting this back, we find that the height is also 64/25, and the dimensions of the box are approximately 12.8 ft by 12.8 ft by 5.12 ft.

A Rectangular Box Is to Have a Square Base and a Volume