1983 AIME Problems/Problem 11
From AoPSWiki
Contents |
Problem
The solid shown has a square base of side length 
. The upper edge is parallel to the base and has length 
. All other edges have length . Given that 
, what is the volume of the solid?
Solution 1
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 
.
Solution 2
Extend 
and 
to meet at 
, and 
and 
to meet at 
. now, we have a regular tetrahedron 
, which has twice the volume of our original solid. This tetrahedron has side length 
. Using the formula for the volume of a regular tetrahedron, which is 
, where S is the side length of the tetrahedron, the volume of our original solid is:
See also
| 1983 AIME (Problems • Resources) | ||
| Preceded by Problem 10 | Followed by Problem 12 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
