Difference between revisions of "1983 AIME Problems/Problem 11"
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Thus, our answer is <math>432-144=288</math>. | Thus, our answer is <math>432-144=288</math>. | ||
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== See also == | == See also == | ||
+ | {{AIME box|year=1983|num-b=10|num-a=12}} | ||
* [[AIME Problems and Solutions]] | * [[AIME Problems and Solutions]] | ||
* [[American Invitational Mathematics Examination]] | * [[American Invitational Mathematics Examination]] |
Revision as of 14:12, 6 May 2007
Problem
The solid shown has a square base of side length . The upper edge is parallel to the base and has length . All edges have length . Given that , what is the volume of the solid?
Solution
First, we find the height of the figure by drawing a perpendicular from the midpoint of to . The hypotenuse of the triangle is the median of equilateral triangle one of the legs is . We apply the pythagorean theorem to find that the height is equal to .
Next, we complete the figure into a triangular prism, and find the area, which is .
Now, we subtract off the two extra pyramids that we included, whose combined area is .
Thus, our answer is .
See also
1983 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |