Let $f$ be the function given by $f(x) = 4x^2 - x^3$, and let $\ell$ be the line $y = 18 - 3x$, where $\ell$ is tangent to the graph of $f$. Let $R$ be the region bounded by the graph of $f$ and the $x$-axis, and let $S$ be the region bounded by the graph of $f$, the line $\ell$, and the $x$-axis.
(a) Show that $\ell$ is tangent to the graph of $y = f(x)$ at the point $x = 3$.
(b) Find the area of $S$.
(c) Find the volume of the solid generated when $R$ is revolved about the $x$-axis.