imc

In order to solve this problem, we must first visualize what the region looks like. We know that, in a three dimensional space, the region consisting of all points within $3$ units of a point would be a sphere with radius $3$. However, we need to find the region containing all points within $3$ units of a segment. It can be seen that our region is a cylinder with two hemispheres/endcaps on either end. We know the volume of our region, so we set up the following equation (the volume of our cylinder + the volume of our two hemispheres will equal $216\pi$): $\frac{4\pi }{3}\cdot 3^3+9\pi x=216\pi$, where $x$ is equal to the length of our line segment. Solving, we find that $x=\boxed{20}$.