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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| + | ==Solution== | ||
| + | |||
| + | 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 | ||
| + | |||
| + | <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> | ||
| + | |||
| + | <math>L = \boxed{(D)=\ 20}</math> | ||
| + | |||
| + | ==Video Solution (HOW TO THINK CREATIVELY!!!)== | ||
| + | https://youtu.be/9D2iKHI3FMk | ||
| + | |||
| + | ~Education, the Study of Everything | ||
| + | |||
| + | ==Video Solution== | ||
| + | https://youtu.be/LGOtggQwSFU | ||
| + | |||
| + | ~Math4All999 | ||
| + | |||
| + | ==See Also== | ||
| + | {{AMC10 box|year=2017|ab=A|num-b=10|num-a=12}} | ||
| + | {{AMC12 box|year=2017|ab=A|num-b=7|num-a=9}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 06:34, 14 September 2024
Contents
Problem
The region consisting of all points in three-dimensional space within 
units of line segment 
has volume 
. What is the length 
?
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
Video Solution (HOW TO THINK CREATIVELY!!!)
~Education, the Study of Everything
Video Solution
~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 | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
