Difference between revisions of "2017 AMC 12A Problems/Problem 8"

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The region consisting of all points in three-dimensional space within <math>3</math> units of line segment <math>\overline{AB}</math> has volume <math>216 \pi</math>. What is the length <math>AB</math>?
 
The region consisting of all points in three-dimensional space within <math>3</math> units of line segment <math>\overline{AB}</math> has volume <math>216 \pi</math>. What is the length <math>AB</math>?
  
<math> \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ 14 </math>
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<math> \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ 24 </math>
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==Video Solution (HOW TO THINK CREATIVELY!!!)==
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https://youtu.be/9D2iKHI3FMk
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~Education, the Study of Everything
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==Solution==
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Let the length <math>AB</math> be <math>L</math>. Then, we see that the region is just the union of the cylinder with central axis <math>\overline{AB}</math> and radius <math>3</math> and the two hemispheres connected to each face of the cylinder (also with radius <math>3</math>). Thus the volume is
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<math>9\pi L + \frac{4}{3}\pi(3)^3 = 9\pi L + 36\pi ( = 216\pi)</math>
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<math>9\pi L = 180\pi</math>
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<math>L = \boxed{(D)=\ 20}</math>
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==See Also==
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{{AMC10 box|year=2017|ab=A|num-b=10|num-a=12}}
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{{AMC12 box|year=2017|ab=A|num-b=7|num-a=9}}
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{{MAA Notice}}

Latest revision as of 14:59, 10 June 2023

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 $AB$?

$\textbf{(A)}\ 6 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ 24$

Video Solution (HOW TO THINK CREATIVELY!!!)

https://youtu.be/9D2iKHI3FMk

~Education, the Study of Everything



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

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

$9\pi L = 180\pi$

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

See Also

2017 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 10
Followed by
Problem 12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 10 Problems and Solutions
2017 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 7
Followed by
Problem 9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

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