Problem $4$: A box with a square base and top and rectangular sides is being constructed out of cardboard. Suppose that

$-$ to build the base and top of the box, it costs $\backslash (\backslash$$ $1\backslash )$ per square inch, and

$-$ to build the other sides of the box, it costs $\backslash (\backslash$$ $0\mathrm{.}50\backslash )$ per square inch.

If you are spending $\backslash (\backslash$$ $54\backslash ),$ find the side lengths $\backslash ($ x $\backslash )$ and $\backslash ($ y $\backslash )$ of the box that would maximize its volume.