Question

What will the dimensions of the resulting cardboard box be if the company wants to maximize the volume and they start with a flat piece of square cardboard 20 feet per side, and then cut smaller squares out of each corner and fold up the sides to create the box?

Answer 1

Suppose that the squares removed from each corner are `x` feet by `x` feet each.

When these are folded up they give a box with a height of `x` feet
and a base of `20 - 2x` feet by `20-2x` feet
for a volume
`V = x(20-2x)^2= 400x -80x^2 + 4x^3`

To find the critical point(s) take the derivative of `V`, set it to zero, and solve for `x`.

`(dV)/(dx) = 400-160x+12x^2`

`=4(3x-10)(x-10) = 0`

Since `x=10` gives a Volume of `0`

`rarr` the critical point for the Volume that is it's maximum occurs when `x=10/3`.

The resulting box will be
`3 1/3 xx 13 1/3 xx 13 1/3` feet