S $89.$ Numerical and Graphical Analysis An open box is to be made from a square piece of material, $36$ inches on a side, by cutting equal squares with sides of length $\backslash ($ x $\backslash )$ from the corners and turning up the sides $($see figure$).$

$($a$)$ Verify that the volume of the box is given by the function

$\backslash [$

V$($x$)=$x$(36-2$ x$)^\{2\}$

$\backslash ]$

$($b$)$ Determine the domain of the function.

$($c$)$ Use a graphing utility to create a table that shows the box height $\backslash ($ x $\backslash )$ and the corresponding volumes $\backslash ($ V $\backslash ).$ Use the table to estimate the dimensions that will produce a maximum volume.

$($d$)$ Use a graphing utility to graph $\backslash ($ V $\backslash )$ and use the graph to estimate the value of $\backslash ($ x $\backslash )$ for which $\backslash ($ V$($x$)\backslash )$ is maximum. Compare your result with that of part $($c$).$