## Task A-3.3.

All points in space whose distance from the line segment $\overline{A B}$ is at most 3 form a solid of volume $216 \pi$. Determine the length of the line segment $\overline{A B}$.
## Solution.

Let $d$ be the length of the line segment $\overline{A B}$.

For each point on that line segment, the set of all points that are at most 3 units away from it is a sphere of radius 3 centered at that point. The union of all these spheres is a body consisting of a cylinder with the centers of its bases at $A$ and $B$, their radii equal to 3, and two hemispheres constructed over the bases of this cylinder (outward).

![](https://cdn.mathpix.com/cropped/2024_05_30_ce71b02a7f068198719dg-18.jpg?height=317&width=928&top_left_y=772&top_left_x=495)

From the conditions of the problem, it follows that

$$
\begin{aligned}
\frac{1}{2} \cdot \frac{4}{3} \cdot 3^{3} \pi+3^{2} \pi d+\frac{1}{2} \cdot \frac{4}{3} \cdot 3^{3} \pi & =216 \pi, & & 1 \text{ point} \\
9 \pi d & =180 \pi, & & 1 \text{ point} \\
d & =20 . & & 1 \text{ point}
\end{aligned}
$$