Question

A closed rectangular box with a square base and a volume of 12 cubic feet is to be constructed using two different types of materials. The base is made of a metal costing $2 per square foot and the remainder is made of a material costing $1 per square foot. Find the dimension of the box for which the cost of materials is minimized.

Answer 1

The dimensions of the box that minimize the cost of materials are x=2 feet and h=3 feet.

Let x be the side length of the square base and let h be the height of the box.

The volume of the box is 12=
`x^(2)` h, so h= 12/
`x^(2)`.

The surface area of the box is 6
`x^(2)` +12x/x=6x+12.

We want to minimize the cost of the materials, which is 2
`x^(2)`+12 (for the base) + 12x+12 (for the rest of the box), which is 14x+2
`x^(2)` +24.

Completing the square, we get 14x+ 2
`x^(2)` +24=2(
`x^(2)`+7x+12)=2(x+3)(x+4)=2(x+4)(x+3).

Since x is the length of a side of the box, it must be positive.

Therefore, we can minimize 2(x+4)(x+3) by minimizing x+3.

The minimum value of x+3 is x+3=0, which occurs when x=−3.

However, this is not a valid solution since the side length of the box cannot be negative.

Therefore, the minimum value of x+3 is x+3=1, which occurs when x=−2.

Since x=−2, we have h=12/(−2)^2 =3.

Therefore, the dimensions of the box that minimize the cost of materials are x=2 feet and h=3 feet.