Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

(→‎Solution)
(→‎Solution)
Line 24: Line 24:
  
 
==Solution==
 
==Solution==
−
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>\dfrac{3}{2}</math>, so the volume of one pyramid is <math>\dfrac{Ah}{3}=5</math>. Thus, the octahedron has volume $2\cdot5=\boxed{(B)}\$
+
The octahedron is just two congruent pyramids glued together by their base. The base of one pyramid is a rhombus with diagonals <math>4</math> and <math>5</math>, for an area <math>A = 10</math>. The height <math>h</math>, of one pyramid, is <math>\dfrac{3}{2}</math>, so the volume of one pyramid is <math>\dfrac{Ah}{3}=5</math>. Thus, the octahedron has volume <math>2\cdot5=\boxed{(B)}\10</math>$
  
 
==See Also==
 
==See Also==

Revision as of 18:54, 3 February 2018

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

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 $2\cdot5=\boxed{(B)}\10$ (Error compiling LaTeX. Unknown error_msg)$

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. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2015_AMC_10B_Problems/Problem_17&oldid=90195"