$1.$ Optimization

Imagine you are constructing a pyramid with square base. This can be visualized by folding up isosceles triangles as pictured below so that their vertices touch at some common point above the center of the square:

Suppose that you only need to construct the surface of a pyramid without the base $($that is$,$ only the triangles pictured above without the square$).$ What is the largest volume possible for a pyramid with square base $($of any dimensions$)$ if you want to use at most $\backslash (100\backslash$mathrm$\{$~m$\}^\{2\}\backslash )$ of material? Show your work carefully, making sure to define your variable$($s$)$ clearly, writing down the intervals of potential values for them, and the function that you are optimizing. $($You may want to look up the formula for the volume of a square pyramid$).$