2017 AMC 10A Problems/Problem 11: Difference between revisions
Created page with "==Problem== The region consisting of all point in three-dimensional space within 3 units of line segment <math>\overline{AB}</math> has volume 216<math>\pi</math>. What is th..." |
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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> | <math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 24</math> | ||
In order to solve this problem, we must first visualize what the region contained looks like. We know that, in a three dimensional plane, the region consisting of all points within <math>3</math> units of a point would be a sphere with radius <math>3</math>. However, we need to find the region containing all points within 3 units of a segment. Therefore, our region is a cylinder with two hemispheres on either end. We know the volume of our region, so we set up the following equation: | |||
<math>\frac{4}{3}\cdot\pi\cdot3^3+9\cdot\pi\cdotx=216</math> | |||
Where <math>x</math> is equal to the length of our line segment. | |||
We isolate <math>x</math>. This comes out to be <math>/boxed{\textbf{(D)}\ 20\qquad}</math>. | |||
Revision as of 14:56, 8 February 2017
Problem
The region consisting of all point in three-dimensional space within 3 units of line segment 
has volume 216
. What is the length 
?
In order to solve this problem, we must first visualize what the region contained looks like. We know that, in a three dimensional plane, the region consisting of all points within 
units of a point would be a sphere with radius . However, we need to find the region containing all points within 3 units of a segment. Therefore, our region is a cylinder with two hemispheres on either end. We know the volume of our region, so we set up the following equation:
$\frac{4}{3}\cdot\pi\cdot3^3+9\cdot\pi\cdotx=216$ (Error compiling LaTeX. Unknown error_msg)
Where 
is equal to the length of our line segment.
We isolate . This comes out to be 
.
