STATEMENT OF PROBLEM

We are given an 8 cm by 10 cm rectangular sheet of material and told to construct an open-topped box of maximum volume.

The simple case is one in which we make square cuts from each corner and fold to make a rectangular box.

Consider the more general case, in which we removecongruent quadrilaterals (wedges) from each corner of the sheet and fold the resulting figure so that the cut edges meet.

How does the maximum volume of the more general box compare with that of the rectangular box?

Hint: Place the sheet in the first quadrant with one vertex at the origin. Draw a line from the origin, at an angle of Pi/4 (toward the point (1, 1)) of length L. Call the endpoint of this line (a, a). Now we will make our cuts from (b, 0) and (0, b) to (a, a), removing a quadrilateral from the corner. We remove corresponding quadrilaterals from the other corners.

Note that if b<a we have a box with a larger top opening than base. We do not have an edge which is perpendicular to the base - unless a = b.