Question

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"?

Answer 1

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 `14`x`72`. It is rather strict assumption and it's not obvious from the problem whether it is allowed. Assume, it is.
Then here are the detailed proof and calculations.

Assuming the dimensions of a cage are `X`, `Y` and `Z`, we can formulate a problem as follows:
Find `X`, `Y` and `Z` such that `V=X*Y*Z` (volume of the cage) reaches its maximum while `S=2*(X*Y+Y*X+Z*X)` (combined area of all 6 sides of a cage) is given. In our case `S=14*72` square inches.

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
`[(X*Y)*(Y*Z)*(Z*X)]^(1/3)<=[(X*Y)+(Y*Z)+(Z*X)]/3`

Using expressions for volume `V` (unknown to be maximized) and area of all sides `S` of a cage (known constant), we can represent the above inequality as
`(V^2)^(1/3)<=S/6` or
`V<=sqrt((S/6)^3)`, where `S=14*72`,
with equality reached only if
`X*Y=Y*Z=Z*X`, that is if
`X=Y=Z`, that is our cage must be a cube.

We have to find an edge of a cube whose side area equals to `14*72`.
Solve the equation:
`6*X^2=14*72` or `X^2=14*12=168`
Therefore, dimensions we are seeking are
`X=Y=Z=sqrt(168)`

This dimension of a side, which is about `12.96`, obviously presents a practical challenge. It will be necessary to cut and attach different pieces of glass to make the cubical cage of this dimension, and it will look like a Tiffany lamp. But beauty is in the eyes of a beholder.
At least, mathematically, it will really be a cage with a maximum volume made from a given piece of glass.