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2017 AMC 12A Problem 8

Problem 8 of 25EasierGeometry

The region consisting of all points in three-dimensional space within 33 units of line segment ABAB has volume 216π.216\pi. What is the length AB?AB?

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Solution

Let h=AB.h=AB. The region is a cylinder of radius 33 and height hh with a hemisphere of radius 33 on each end. The cylinder has volume π⋅32⋅h=9πh,\pi\cdot3^2\cdot h=9\pi h, and the two hemispheres together form a sphere of volume 43π⋅33=36π.\dfrac{4}{3}\pi\cdot3^3=36\pi. So 9πh+36π=216π, 9\pi h+36\pi=216\pi, giving h=20.h=20. Thus, the correct answer is D.

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Concepts: cylinder · sphere · volume

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.