A sheet of cardboard, 12 X 12 inches, is used to make an open box by cutting squares of equal size from the four corners and folding up the sides. What size squares should be cut to obtain a box with the largest possible volume?

Question

A sheet of cardboard, 12  X 12 inches, is used to make an open box by cutting squares of equal size from the four corners and folding up the sides.  What size squares should be cut to obtain a box with the largest possible volume?

anonymous · Answer

In general, largest volume for limited surface area comes from figures that are nearly spheres, or here, cubic.

Open box described is made by taking sides one-third of original square, so sides are 4 inches on edge.

You can set this up as an optimization problem, which is to maximize length x width x height subject to the limits of the amount of material you have. Better math approach, but if you try my first suggestion, you can compare the 4-inch cube against the rectangular solids made with slightly larger or smaller squares.