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)}\ | + | <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!!!)== | ||
+ | https://youtu.be/9D2iKHI3FMk | ||
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+ | ~Education, the Study of Everything | ||
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+ | |||
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==Solution== | ==Solution== |
Latest revision as of 14:59, 10 June 2023
Problem
The region consisting of all points in three-dimensional space within units of line segment
has volume
. What is the length
?
Video Solution (HOW TO THINK CREATIVELY!!!)
~Education, the Study of Everything
Solution
Let the length be
. Then, we see that the region is just the union of the cylinder with central axis
and radius
and the two hemispheres connected to each face of the cylinder (also with radius
). Thus the volume is
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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