2017 AMC 12A Problems/Problem 8

Problem

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 $AB$ ?

$\textbf{(A)}\ 6 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ 14$

Solution

Let the length $AB$ be $L$ . Then, we see that the region is just the union of the cylinder with central axis $\overline{AB}$ and radius $3$ and the two hemispheres connected to each face of the cylinder (also with radius $3$ ). Thus the volume is

$9\pi L + \frac{4}{3}\pi(3)^3 = 9\pi L + 36\pi ( = 216\pi)$

$9\pi L = 180\pi$

$L = \boxed{(D)=\ 20}$

2017 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
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2017 AMC 12A Problems/Problem 8

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