$($BC only, calculator$)$

Consider the region bounded be the $y-$axis, $y=10,$ and $y=1+6x^{\frac{3}{2}}.$

$($a$)$ Set up$,$ but do not evaluate, an integral equation that will find the value of $k$ so that $x=k$ cuts the region into $2$ parts of equal area.

$($b$)$ Find the length of the curve $y=1+6x^{\frac{3}{2}}$ on the interval $0,1.$

$($c$)$ The region is the base of a solid. For this solid, the crosssections perpendicular to the $x-$axis are rectangles with a height of $3$ times that of its width. Find the volume of this solid.