The region consisting of all points in three-dimensional space within $3$ units of line segment $\overline{AB}$ has volume $216 \pi$. What is the length $AB$?
2026-04-02 16:54:33.1775148873
Length of $\overline{AB}$ in this problem
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Such region is given by the union of a cylinder and two half-spheres, whose volume is given by $$ \pi 3^2 \ell + \frac{4}{3}\pi 3^3 = 216\pi $$ from which $\ell=20$.