Difference between revisions of "2017 AMC 10A Problems/Problem 11"

(Problem)
(Problem)
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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:
+
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 (the volume of our cylinder + the volume of our two hemispheres will equal <math>216 \pi</math>):
  
<math>\frac{4\pi}{3}3^3+9\pix=216</math>
+
<math>\frac{4 \pi }{3}3^3+9 \pi x=216</math>
  
 
Where <math>x</math> is equal to the length of our line segment.
 
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}</math>
 
We isolate <math>x</math>.  This comes out to be <math>\boxed{\textbf{(D)}\ 20}</math>

Revision as of 16:27, 8 February 2017

Problem

The region consisting of all point in three-dimensional space within 3 units of line segment $\overline{AB}$ has volume 216$\pi$. What is the length $\textit{AB}$?

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

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 $3$ units of a point would be a sphere with radius $3$. 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 (the volume of our cylinder + the volume of our two hemispheres will equal $216 \pi$):

$\frac{4 \pi }{3}3^3+9 \pi x=216$

Where $x$ is equal to the length of our line segment.

We isolate $x$. This comes out to be $\boxed{\textbf{(D)}\ 20}$

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