https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Laura.yingyue.zhang&feedformat=atom AoPS Wiki - User contributions [en] 2021-07-26T06:18:32Z User contributions MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_10A_Problems/Problem_11&diff=90049 2017 AMC 10A Problems/Problem 11 2018-01-29T13:26:03Z <p>Laura.yingyue.zhang: </p> <hr /> <div>==Problem==<br /> <br /> The region consisting of all points in three-dimensional space within 3 units of line segment &lt;math&gt;\overline{AB}&lt;/math&gt; has volume 216&lt;math&gt;\pi&lt;/math&gt;. What is the length &lt;math&gt;\textit{AB}&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A)}\ 6\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 24&lt;/math&gt;<br /> <br /> ==Solution==<br /> 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 &lt;math&gt;3&lt;/math&gt; units of a point would be a sphere with radius &lt;math&gt;3&lt;/math&gt;. However, we need to find the region containing all points within 3 units of a segment. It can be seen that 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 &lt;math&gt;216 \pi&lt;/math&gt;):<br /> <br /> &lt;math&gt;\frac{4 \pi }{3} \cdot 3^3+9 \pi x=216 \pi&lt;/math&gt;, where &lt;math&gt;x&lt;/math&gt; is equal to the length of our line segment.<br /> <br /> Solving, we find that &lt;math&gt;x = \boxed{\textbf{(D)}\ 20}&lt;/math&gt;.<br /> <br /> ==Diagram==<br /> <br /> http://i.imgur.com/cwNt293.png<br /> <br /> ==See Also==<br /> {{AMC10 box|year=2017|ab=A|num-b=10|num-a=12}}<br /> {{MAA Notice}}</div> Laura.yingyue.zhang