Question

A box is to be made out of a 8 by 16 piece of cardboard. Squares of equal size will be cut out of each corner, and then the ends and sides will be folded up to form a box with an open top. Find the length L, width W, and height H of the resulting box that maximizes the volume. (Assume that W≤L).

Answer 1

Let a be the side length of the cut-out squares. Then the volume will be:
V = (16-2a)(8-2a)*a = 4a^3 - 48a^2 + 128a
Differentiating with respect to a gives:
12a^2 - 96a - 128
Put the first differential = 0 and solve for a to find the maximum volume.
This gives a = 1.6
L = 12.8, W = 4.8 and H = 1.6

Answer 2

krop, those numbers are wrong

Answer 3

Sorry I made a calculation error. The last two lines should read:
This gives a = 1.7
L = 12.6, W = 4.6 and H = 1.7