Difference between revisions of "2008 AMC 12A Problems/Problem 18"
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− | [[WLOG]], we let <math>AB</math> go between the <math>x</math> and <math>y</math> axes, <math>BC</math> between <math>y</math> and <math>z</math> axes | + | [[WLOG]], we let <math>AB</math> go between the <math>x</math> and <math>y</math> axes, <math>BC</math> between <math>y</math> and <math>z</math> axes, <math>CA</math> between <math>z</math> and <math>x</math> axes. Let <math>x,y,z</math> be <math>OA,OB,OC,</math> respectively. By the [[Pythagorean Theorem]], <math>x^2+y^2=25</math>, <math>y^2+z^2=36</math>, <math>z^2+x^2=49</math>. Thus, <math>x^2 = 30</math>, <math>y^2 = 19</math>, and <math>z^2 = 6</math>. Thus the volume of the tetraehdron is <math>\frac{\sqrt{30\cdot 19\cdot 6}}{6}=\sqrt{95}\Rightarrow \boxed{C}</math>. |
Revision as of 20:47, 17 February 2008
Problem
A triangle with sides
,
,
is placed in the three-dimensional plane with one vertex on the positive
axis, one on the positive
axis, and one on the positive
axis. Let
be the origin. What is the volume if
?
Solution
WLOG, we let go between the
and
axes,
between
and
axes,
between
and
axes. Let
be
respectively. By the Pythagorean Theorem,
,
,
. Thus,
,
, and
. Thus the volume of the tetraehdron is
.
2008 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 14 |
Followed by Problem 16 |
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 12 Problems and Solutions |