Difference between revisions of "2008 AMC 12B Problems/Problem 18"
(New page: ==Problem== A pyramid has a square base <math>ABCD</math> and vertex <math>E</math>. The area of square <math>ABCD</math> is <math>196</math>, and the areas of <math>\triangle ABE</math> ...) |
(→Solution) |
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Line 8: | Line 8: | ||
<math>13^2-(14-a)^2=h^2 \\ | <math>13^2-(14-a)^2=h^2 \\ | ||
− | 15^2-a^2=h^2</math> | + | 15^2-a^2=h^2</math> |
Setting them equal to each other and simplifying gives | Setting them equal to each other and simplifying gives |
Revision as of 21:45, 30 November 2008
Problem
A pyramid has a square base and vertex
. The area of square
is
, and the areas of
and
are
and
, respectively. What is the volume of the pyramid?
Solution
Let be the height of the pyramid and
be the distance from
to
. The side length of the base is 14. The side lengths of
and
are
and
, respectively. We have a systems of equations through the Pythagorean Theorem:
Setting them equal to each other and simplifying gives
.
Therefore, , and the volume of the pyramid is
.