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Difference between revisions of "2015 AMC 10B Problems/Problem 17"
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label("5",(5,4,0)--(0,4,0),SE);
 
label("5",(5,4,0)--(0,4,0),SE);
 
</asy>
 
</asy>
−
<math>$\textbf{(A) } \dfrac{75}{12}
+
 
 +
<math>
 +
\textbf{(A) } \dfrac{75}{12}
 
\qquad\textbf{(B) } 10
 
\qquad\textbf{(B) } 10
 
\qquad\textbf{(C) } 12
 
\qquad\textbf{(C) } 12
Line 20: Line 22:
 
\qquad\textbf{(E) } 15
 
\qquad\textbf{(E) } 15
 
</math>
 
</math>
 +
 +
==Solution==
 +
The octahedron is just two congruent pyramids glued together. The base of one of the pyramids is a rhombus with diagonals <math>4</math> and <math>5</math>, for an area of <math>10</math>. The height of one of the pyramids is then <math>\dfrac{3}{2}</math>, so the volume is <math>\dfrac{Bh}{3}=5</math>. Thus, the octahedron has volume <math>\boxed{\mathbf{(B)}\ 10}</math>
 +
 +
==See Also==
 +
{{AMC10 box|year=2015|ab=B|num-b=16|num-a=18}}
 +
{{MAA Notice}}

Revision as of 13:55, 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]

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

Solution

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

See Also

2015 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

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