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Difference between revisions of "1992 AIME Problems/Problem 7"
 
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== Problem ==
 
== Problem ==
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Faces <math>ABC^{}_{}</math> and <math>BCD^{}_{}</math> of tetrahedron <math>ABCD^{}_{}</math> meet at an angle of <math>30^\circ</math>. The area of face <math>ABC^{}_{}</math> is <math>120^{}_{}</math>, the area of face <math>BCD^{}_{}</math> is <math>80^{}_{}</math>, and <math>BC=10^{}_{}</math>. Find the volume of the tetrahedron.
  
 
== Solution ==
 
== Solution ==
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Since the area <math>BCD=80=\frac{1}{2}\cdot10\cdot16</math>, the perpendicular from <math>D</math> to <math>BC</math> has length <math>16</math>.
 +
 +
The perpendicular from <math>D</math> to <math>ABC</math> is <math>16 \cdot \sin 30^\circ=8</math>. Therefore, the volume is <math>\frac{8\cdot120}{3}=\boxed{320}</math>.
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==Solution 2==
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The area of <math>ABC</math> is 120 and <math>BC</math>=10, the slant height is 24. Height from <math>A</math> to <math>BCD</math> is <math>24 \cdot \sin 30^\circ=12</math>. Since area of <math>BCD</math> is 80, the volume of tetrahedron <math>ABCD</math>= <math>\frac{80\cdot12}{3}=\boxed{320}</math>.
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== See also ==
 
== See also ==
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* [[1992 AIME Problems]]
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{{AIME box|year=1992|num-b=6|num-a=8}}
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[[Category:Intermediate Geometry Problems]]
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{{MAA Notice}}

Latest revision as of 03:29, 3 February 2025

Problem

Faces $ABC^{}_{}$ and $BCD^{}_{}$ of tetrahedron $ABCD^{}_{}$ meet at an angle of $30^\circ$. The area of face $ABC^{}_{}$ is $120^{}_{}$, the area of face $BCD^{}_{}$ is $80^{}_{}$, and $BC=10^{}_{}$. Find the volume of the tetrahedron.

Solution

Since the area $BCD=80=\frac{1}{2}\cdot10\cdot16$, the perpendicular from $D$ to $BC$ has length $16$.

The perpendicular from $D$ to $ABC$ is $16 \cdot \sin 30^\circ=8$. Therefore, the volume is $\frac{8\cdot120}{3}=\boxed{320}$.

Solution 2

The area of $ABC$ is 120 and $BC$=10, the slant height is 24. Height from $A$ to $BCD$ is $24 \cdot \sin 30^\circ=12$. Since area of $BCD$ is 80, the volume of tetrahedron $ABCD$= $\frac{80\cdot12}{3}=\boxed{320}$.


See also

1992 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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