Let $R$ be the region in the first quadrant bounded by the graph of $y = 8 - x^{\frac{3}{2}}$, the $x$-axis, and the $y$-axis.
(a) Find the area of the region $R$.
(b) Find the volume of the solid generated when $R$ is revolved about the $x$-axis.
(c) The vertical line $x = k$ divides the region $R$ into two regions such that when these two regions are revolved about the $x$-axis, they generate solids with equal volumes. Find the value of $k$.