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

The region is a cylinder of radius

3

with a hemisphere capping each end

The two hemispheres combine into a full sphere:

9\pi h+\dfrac{4}{3}\pi 3^3=216\pi

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.

9\pi h+36\pi=216\pi,

giving

h=20.

Thus, the correct answer is D.

Problem 8 in other years