Applied Optimization

Question

According to postal regulations, a carton is classified as "oversized" if the sum of its height and girth (the perimeter of its base) exceeds 156 in. Find the dimensions of a carton (in inches) with square base that is not oversized and has maximum volume.

Answer 1

Let x = the side length of the box's square base and h = the box's height. The constraint is that the girth (perimeter of base) plus the height must be 156 inches:

4x + h = 156

h = 156 - 4x

The box's volume is:

V = x²·h = x²·(156-4x) = -4x³ + 156x²

To find the max volume, take the derivative of V wrt x, set it to zero and solve for x:

dV/dx = -12x² + 312x

0 = -12x(x-26)

x = 0 and 26

We can't have a box with a side length of zero, so x = 26 in. h = 156 - 4x = 156 - 104 = 52 in.