Difference between revisions of "2008 AMC 12B Problems/Problem 18"
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==Solution== | ==Solution== | ||
− | Let <math>h</math> be the height of the pyramid and <math>a</math> be the distance from <math>h</math> to <math>CD</math>. The side length of the base is 14. The | + | Let <math>h</math> be the height of the pyramid and <math>a</math> be the distance from <math>h</math> to <math>CD</math>. The side length of the base is <math>14</math>. The heights of <math>\triangle ABE</math> and <math>\triangle CDE</math> are <math>2\cdot105\div14=15</math> and <math>2\cdot91\div14=13</math>, respectively. Consider a side view of the pyramid from <math>\triangle BCE</math>. We have a systems of equations through the Pythagorean Theorem: |
<math>13^2-(14-a)^2=h^2 \\ | <math>13^2-(14-a)^2=h^2 \\ | ||
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Therefore, <math>h=\sqrt{15^2-9^2}=12</math>, and the volume of the pyramid is <math>\frac{bh}{3}=\frac{12\cdot 196}{3}=\boxed{784 \Rightarrow E}</math>. | Therefore, <math>h=\sqrt{15^2-9^2}=12</math>, and the volume of the pyramid is <math>\frac{bh}{3}=\frac{12\cdot 196}{3}=\boxed{784 \Rightarrow E}</math>. | ||
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==See also== | ==See also== | ||
{{AMC12 box|year=2008|ab=B|num-b=17|num-a=19}} | {{AMC12 box|year=2008|ab=B|num-b=17|num-a=19}} |
Latest revision as of 12:01, 2 February 2015
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 . The heights of and are and , respectively. Consider a side view of the pyramid from . 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 .
See also
2008 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 17 |
Followed by Problem 19 |
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 |
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