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Difference between revisions of "2015 AMC 10B Problems/Problem 17"
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==Problem==
 
==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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The centers of the faces of the right rectangular prism shown below are joined to create an octahedron. What is the volume of this octahedron?
  
 
<asy>
 
<asy>
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</math>
 
</math>
  
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==Solution==
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==Solution 1==
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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>
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The octahedron is just two congruent pyramids glued together by their base. The base of one pyramid is a rhombus with diagonals <math>4</math> and <math>5</math>, for an area <math>A=10</math>. The height <math>h</math>, of one pyramid, is <math>\frac{3}{2}</math>, so the volume of one pyramid is <math>\frac{Ah}{3}=5</math>. Thus, the octahedron has volume <math>2\cdot5=\boxed{\textbf{(B) }10}</math>
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 +
==Solution 2 (Scaling)==
 +
The "base" of the octahedron is half the base of the rectangular prism because it is connected by the midpoints. Additionally, the volume of an octahedron is <math>\dfrac{1}{3}</math> of its respective prism. Thus, the octahedron's volume is <math>\dfrac{1}{2} \cdot \dfrac{1}{3} = \dfrac{1}{6}</math> of the rectangular prism's volume, meaning that the answer is <math>3 \cdot 4 \cdot 5 \cdot \dfrac{1}{6} = \boxed{\\10}</math>
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 +
==Video Solution==
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https://www.youtube.com/watch?v=NEQNfr_T8OA
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2015|ab=B|num-b=16|num-a=18}}
 
{{AMC10 box|year=2015|ab=B|num-b=16|num-a=18}}
 
{{MAA Notice}}
 
{{MAA Notice}}
 +
 +
[[Category: Introductory Geometry Problems]]
 +
[[Category: 3D Geometry Problems]]

Latest revision as of 19:57, 25 August 2023

Problem

The centers of the faces of the right rectangular prism shown below are joined to create an octahedron. What is the volume of this 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 1

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 $\frac{3}{2}$, so the volume of one pyramid is $\frac{Ah}{3}=5$. Thus, the octahedron has volume $2\cdot5=\boxed{\textbf{(B) }10}$

Solution 2 (Scaling)

The "base" of the octahedron is half the base of the rectangular prism because it is connected by the midpoints. Additionally, the volume of an octahedron is $\dfrac{1}{3}$ of its respective prism. Thus, the octahedron's volume is $\dfrac{1}{2} \cdot \dfrac{1}{3} = \dfrac{1}{6}$ of the rectangular prism's volume, meaning that the answer is $3 \cdot 4 \cdot 5 \cdot \dfrac{1}{6} = \boxed{\\10}$

Video Solution

https://www.youtube.com/watch?v=NEQNfr_T8OA

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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