A closed cardboard box is made with a square top and bottom, and a square horizontal shelf inside that divides the interior in half. A total of 12 square meters of cardboard is used to make the top, sides, bottom, and shelf of the box. What should the dimensions of the box be to maximize its volume?

Question

length = m.
width = m.
total height = m.

Answer 1

So, we have 3 square sections and 4 sides. If the sides are of length x, and the height is y, then the cardboard used is

3x^2 + 4xy = 12

so, y = (12-3x^2)/4x

The volume is

v = x^2y = x^2(12-3x^2)/4x
= (12x - 3x^3)/4

dv/dx = 3 - 9/4 x^2
max volume when dv/dx=0, or x=2/√3

Now just figure y.

Answer 2

12