Difference between revisions of "2015 AMC 10B Problems/Problem 17"

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==Problem==
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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?
  
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<asy>
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import three; size(2inch);
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currentprojection=orthographic(4,2,2);
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draw((0,0,0)--(0,0,3),dashed);
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draw((0,0,0)--(0,4,0),dashed);
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draw((0,0,0)--(5,0,0),dashed);
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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));
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draw((0,4,3)--(5,4,3)--(5,4,0));
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label("3",(5,0,3)--(5,0,0),W);
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label("4",(5,0,0)--(5,4,0),S);
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label("5",(5,4,0)--(0,4,0),SE);
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</asy>
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$\textbf{(A) } \dfrac{75}{12}
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\qquad\textbf{(B) } 10
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\qquad\textbf{(C) } 12
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\qquad\textbf{(D) } 10\sqrt2
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\qquad\textbf{(E) } 15
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$

Revision as of 13:46, 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]

$\textbf{(A) } \dfrac{75}{12} \qquad\textbf{(B) } 10 \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 10\sqrt2 \qquad\textbf{(E) } 15 $