Difference between revisions of "2015 AMC 10B Problems/Problem 17"
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==Solution 2 (Scaling)== | ==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> | 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> | ||
| + | |||
| + | ==Video Solution== | ||
| + | https://youtu.be/0uCMSH7-Ubk | ||
==See Also== | ==See Also== | ||
Revision as of 19:56, 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]](http://latex.artofproblemsolving.com/0/f/e/0fecfb360827e29c3e9ad69126c0215a16cfd2ab.png)
Solution 1
The octahedron is just two congruent pyramids glued together by their base. The base of one pyramid is a rhombus with diagonals 
and 
, for an area 
. The height 
, of one pyramid, is 
, so the volume of one pyramid is 
. Thus, the octahedron has volume 
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 
of its respective prism. Thus, the octahedron's volume is 
of the rectangular prism's volume, meaning that the answer is 
Video Solution
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
