2017 AMC 12A Problem 8

Attempt Problem 8 of the 2017 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2017 AMC 12A solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

The region consisting of all points in three-dimensional space within $3$ units of line segment $AB$ has volume $216\pi.$ What is the length $AB?$

$6$

$12$

$18$

$20$

$24$

Answer: D

Concepts:cylinder sphere volume

Difficulty rating: 1440

Solution:

Let $h=AB.$ The region is a cylinder of radius $3$ and height $h$ with a hemisphere of radius $3$ on each end.

The cylinder has volume $\pi\cdot3^2\cdot h=9\pi h,$ and the two hemispheres together form a sphere of volume $\dfrac{4}{3}\pi\cdot3^3=36\pi.$ So $9\pi h+36\pi=216\pi,$ giving $h=20.$

Thus, the correct answer is D.

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