BOXES

Just about every calculus student has solved the problem:

Suppose you are to construct a box from a rectangular sheet with dimensions by cutting the corners and folding up the sides. How should you cut the corners so that the volume of the box that results is a maximum?

Here’s a related problem for you calculus III students out there—those of you who think you are up to the challenge.

Suppose you are to construct a box from a sheet with dimensions by cutting the corners and folding up the sides. In order not to waste any material, you will make boxes in the same manner from the leftover corners. You will continue to make boxes from the leftovers ad infinitum. How should you cut the corners so that the total volume of the boxes that result is a maximum?

For those of you who have not reached calculus III yet, try solving this problem by assuming that the ratio of the corner length to the length of the sheet is constant at each level. (In fact, those of you who have taken calculus III may want to assume this as well.)

(Note: There is an article in the September, 2003 issue of The College Mathematics Journal titled Folding boxes, perhaps for fun and profit you might like to peruse.)