Question $10$: $(10$ points$)$

A company wishes to produce a rectangular box with volume $\backslash (1\backslash$mathrm$\{$~m$\}^\{3\}\backslash ).$ Suppose that the length, breadth and height in meters of the box are given by $\backslash ($ x$,$ y $\backslash )$ and $\backslash ($ z $\backslash )$ respectively, and that the cost of materials is $\backslash (4\backslash )$ per $\backslash ($ m$^\{2\}\backslash )$ for the bottom and $3$ per $\backslash ($ m$^\{2\}\backslash )$ for the sides. The cost for the top with dimension $\backslash ($ x $\backslash )$ and $\backslash ($ y $\backslash )$ can be neglected $($i$.$e$.,$ it is assumed to be $\backslash (0\backslash )).$ Express the cost of the box as a function $\backslash ($ C$($x$,$ y$)\backslash )$ of the variables x and y only.

$\backslash [$

C$($x$,$ y$)=$

$\backslash ]$

Find the dimensions that minimise the cost.

The optimal dimensions are $\backslash (($x$,$ y$,$ z$)=(\backslash )$

Preferably leave your answer in terms of fractions, roots and powers. If you must evaluate it then you must give your answer with $3$ digits after the decimal point.