Eighteen ounces of Salubrious Cereal is packaged in a box (inches) × × 11". The box's volume is about 180 cubic inches; its total surface area, A, is about 249 square inches. Could the manufacturer keep the same volume but reduce A by changing the dimensions of the package? If so, the company could save money.

Consider 3" × 6" × 10" and 4" × 5" × 9" boxes. Their volume is 180 cubic inches, but A = 216 and 202 square inches, respectively. Clearly, for a fixed volume, surface area can vary, suggesting that a certain shape might produce a minimum surface area.

The problem can be rephrased this way: Let x, y, and z be the sides of a box such that xyz = V for a fixed V. Find the minimum value of the function A(x,y,z) = 2xy + 2xz + 2yz. This looks complicated. An appropriate strategy would be to first solve a simpler two-dimensional problem in order to develop methods for...