A rectangular box has height $hcm,$ width $xcm$ and length $2xcm.$ It is designed to have a volume equal to $1$ lime $(1000c{m}^3).(15$ points$)$

$($a$)$ Show that $h=\frac{500}{x^2}cm$

$($b$)$ Find an expression for the total surface area, $s$ in $c{m}^2,$ of the box in terms of $x.$

$($c$)$ What dimensions of the box will produce the minimum surface area?