Find the dimensions of the box that minimize cost.

Question

You are constructing a box for your cat to sleep in. The plush material for the square bottom of the box costs $5/ft² and the material for the sides costs $2/ft². You need a box with volume 4 ft². Find the dimensions of the box that minimize cost.

Any help would be amazing thanks

Answer 1

The material you are purchasing is based on area (so many dollars per square foot) so you need an equation for area, i.e., surface area.

We'll assume there is no top on the box. Also, notice that the bottom is a square (so L x L)

The surface area of the bottom of the box is one side times the other (L•L)

The surface area of the two ends is width x height (L•H)

The surface area of the two sides is length x height (L•H)

So the total surface area is L•L + 4(L•H)

You also know that you are limited to a volume of 4 cubic feet so L•L•H = 4 or H = 4/L²

So SA = L•L + 4L•(4/L²)

And since the cost for the bottom is $5 and the cost for the sides is $2 we can say:

Cost = 5•L•L + 2•4L•(4/L²)

Simplifying:

C(L) = 5L² + 32/L

C(L) = 5L² + 32L^-1

To minimize cost, take the derivative and set it equal to zero:

C'(L) = 10L + -32L^-2

0 = 10L + -32L^-2

0 = 10L + -32/L²

32/L² = 10L

32 = 10L³

32/10 = L³

16/5 = L³

L = cuberoot(16/5) = 1.474 ft