Answer
$18 \pi $ or $ \approx 56.549 \text{ cubic feet per foot} $
Work Step by Step
Let us say that the tangent Line contains the points $(x, c)$. The slope the tangent Line to the graph of $f(x)$ at $(x, c)$ can be written as $f'(x) =\lim\limits_{x \to c} \dfrac{f(x)-f(c)}{x-c} ...(1)$
In order to find the rate of change of volume with radius, we will use equation (1).
Plug in the given data to obtain:
$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 \text{ cubic feet per foot} $
