# What dimensions would you need to make a glass cage with maximum volume if you have a piece of glass that is 14" by 72"?

##### 1 Answer

If we are allowed to cut and attach to each other pieces of glass of any size, we have to make a cubical cage with the area of all sides combined equal to the area of a piece of glass given, that is

Then here are the detailed proof and calculations.

Assuming the dimensions of a cage are

Find

As we noted above, we assume that all glass can be used to make a cage without any difficulties of cutting and attaching pieces together.

As we know, arithmetic average of a group of numbers is greater than or equal to their geometric average with equality reached only if all numbers are equal to each other.

Therefore, we can state that

Using expressions for volume

with equality reached only if

We have to find an edge of a cube whose side area equals to

Solve the equation:

Therefore, dimensions we are seeking are

This dimension of a side, which is about

At least, mathematically, it will really be a cage with a maximum volume made from a given piece of glass.