Answer
$18 \pi$ or $ \approx 56.549$ cubic feet per foot
Work Step by Step
Consider the tangent line that contains the point $(x, c)$. The slope the tangent line to the graph of $f(x)$ at $(x, c)$ can be written as
$f'(x) =\lim\limits_{x \to c} \dfrac{f(x)-f(c)}{x-c}$
We use this equation to find the rate of change of volume with radius.
Consider $h=3$ and $V(r)=3\pi r^2$. Plugging into the above equation gives us:
$V'(3) =\lim\limits_{r \to 3} \dfrac{V(r)-V(3)}{r-3} \\=\lim\limits_{r \to 3} \dfrac{3\pi r^2-(3 \pi)(3^2)}{r-3}\\=\lim\limits_{r \to 3} \dfrac{ 3\pi (r^2-9) }{r-3}\\=\lim\limits_{r \to 3} \dfrac{ 3\pi (r-3)(r+3) }{r-3}\\=\lim\limits_{r \to 3} 3 \pi (r+3) \\=18 \pi \\ \approx 56.549~ft^3/ft$
