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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>?
  
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<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>
  
 
==Solution==
 
==Solution==
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<math>L = \boxed{(D)=\ 20}</math>
 
<math>L = \boxed{(D)=\ 20}</math>
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 +
==Video Solution (HOW TO THINK CREATIVELY!!!)==
 +
https://youtu.be/9D2iKHI3FMk
 +
 +
~Education, the Study of Everything
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 +
==Video Solution==
 +
https://youtu.be/LGOtggQwSFU
 +
 +
~Math4All999
  
 
==See Also==
 
==See Also==

Latest revision as of 06:34, 14 September 2024

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 $AB$?

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

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

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

$9\pi L = 180\pi$

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

Video Solution (HOW TO THINK CREATIVELY!!!)

https://youtu.be/9D2iKHI3FMk

~Education, the Study of Everything

Video Solution

https://youtu.be/LGOtggQwSFU

~Math4All999

See Also

2017 AMC 10A (Problems • Answer Key • Resources)
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 (Problems • Answer Key • Resources)
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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