Difference between revisions of "2015 AMC 10B Problems/Problem 17"
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| + | ==Problem== | ||
| + | The centers of the faces of the right rectangular prism shown below are joined to create an octahedron, What is the volume of the octahedron? | ||
| + | <asy> | ||
| + | import three; size(2inch); | ||
| + | currentprojection=orthographic(4,2,2); | ||
| + | draw((0,0,0)--(0,0,3),dashed); | ||
| + | draw((0,0,0)--(0,4,0),dashed); | ||
| + | draw((0,0,0)--(5,0,0),dashed); | ||
| + | draw((5,4,3)--(5,0,3)--(5,0,0)--(5,4,0)--(0,4,0)--(0,4,3)--(0,0,3)--(5,0,3)); | ||
| + | draw((0,4,3)--(5,4,3)--(5,4,0)); | ||
| + | label("3",(5,0,3)--(5,0,0),W); | ||
| + | label("4",(5,0,0)--(5,4,0),S); | ||
| + | label("5",(5,4,0)--(0,4,0),SE); | ||
| + | </asy> | ||
| + | |||
| + | $\textbf{(A) } \dfrac{75}{12} | ||
| + | \qquad\textbf{(B) } 10 | ||
| + | \qquad\textbf{(C) } 12 | ||
| + | \qquad\textbf{(D) } 10\sqrt2 | ||
| + | \qquad\textbf{(E) } 15 | ||
| + | $ | ||
Revision as of 14:46, 6 March 2015
Problem
The centers of the faces of the right rectangular prism shown below are joined to create an octahedron, What is the volume of the octahedron?
![[asy] import three; size(2inch); currentprojection=orthographic(4,2,2); draw((0,0,0)--(0,0,3),dashed); draw((0,0,0)--(0,4,0),dashed); draw((0,0,0)--(5,0,0),dashed); draw((5,4,3)--(5,0,3)--(5,0,0)--(5,4,0)--(0,4,0)--(0,4,3)--(0,0,3)--(5,0,3)); draw((0,4,3)--(5,4,3)--(5,4,0)); label("3",(5,0,3)--(5,0,0),W); label("4",(5,0,0)--(5,4,0),S); label("5",(5,4,0)--(0,4,0),SE); [/asy]](http://latex.artofproblemsolving.com/0/f/e/0fecfb360827e29c3e9ad69126c0215a16cfd2ab.png)
$\textbf{(A) } \dfrac{75}{12} \qquad\textbf{(B) } 10 \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 10\sqrt2 \qquad\textbf{(E) } 15 $
