Art of Problem Solving
Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2017 AMC 10A Problems/Problem 11: Difference between revisions

← Older editNewer edit →
Dhman2727 (talk | contribs)
16 edits
Dhman2727 (talk | contribs)
16 edits
Line 7: Line 7:
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:


<math>\frac{4\pi}{3}\cdot{3^3}+9\pi\cdotx=216</math>
<math>\frac{4\pi}{3}3^3+9\pix=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\qquad}</math>.
We isolate <math>x</math>.  This comes out to be <math>\boxed{\textbf{(D)}\ 20}</math>

Revision as of 14:58, 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:

$\frac{4\pi}{3}3^3+9\pix=216$ (Error compiling LaTeX. Unknown error_msg)

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

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

Retrieved from "https://wiki.aopstest.com/wiki/index.php?title=2017_AMC_10A_Problems/Problem_11&oldid=82840"