Suppose you have 144 square inches of cardboard. Your task is to build a box with a square bottom and no top. find the dimensions of the box to maximize volume.

Question

Suppose you have 144 square inches of cardboard. Your task is to build a box with a square bottom and no top. find the dimensions of the box to maximize volume.

Answer 1

We will need to cut it into 5 pieces to make 4 sides and a bottom.

 

Since the bottom must be a square, the sides will also be squares, and the unfolded box will look like a plus sign.

 

Thus, we will need 5 pieces from a piece 144in², and each piece will have an area of 28.8in².

 

As a result, the sides will have a length equal to the square root of 28.8, which is ≅ 5.36in.

 

The box will then consist of a square bottom with an area of 28.8 in², and sides of 5.36in.  Those dimensions will also apply to the sides.

 

The length, width, and height of the box will be 5.36in, so the volume will be 5.36x5.36x5.36 ≅ 154in³.