2015 AMC 10B Problems/Problem 17

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?

$[asy] import three; size(2inch); currentprojection=orthographic(4,2,2); draw((0,0,0)--(0,0,3),dashed); draw((0,0,0)--(0,4,0),dashed); draw((0,0,0)--(5,0,0),dashed); draw((5,4,3)--(5,0,3)--(5,0,0)--(5,4,0)--(0,4,0)--(0,4,3)--(0,0,3)--(5,0,3)); draw((0,4,3)--(5,4,3)--(5,4,0)); label("3",(5,0,3)--(5,0,0),W); label("4",(5,0,0)--(5,4,0),S); label("5",(5,4,0)--(0,4,0),SE); [/asy]$

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

Solution

The octahedron is just two congruent pyramids glued together by their base. The base of one pyramid is a rhombus with diagonals $4$ and $5$ , for an area $A = 10$ . The height $h$ , of one pyramid, is $\dfrac{3}{2}$ , so the volume of one pyramid is $\dfrac{Ah}{3}=5$ . Thus, the octahedron has volume $2\cdot5=\boxed{\mathbf{(B)}\ 10}$

2015 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 16	Followed by Problem 18
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2015 AMC 10B Problems/Problem 17

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