Difference between revisions of "1983 AIME Problems/Problem 11"
I like pie (talk | contribs) m |
I like pie (talk | contribs) (Undo revision by I like pie (Talk)) |
||
| Line 4: | Line 4: | ||
</asy></center> -->[[Category:Asymptote needed]] | </asy></center> -->[[Category:Asymptote needed]] | ||
| − | |||
[[Image:1983Number11.JPG]] | [[Image:1983Number11.JPG]] | ||
== Solution == | == Solution == | ||
Revision as of 11:14, 25 April 2008
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 | ||
