Difference between revisions of "2000 AIME I Problems/Problem 8"
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Revision as of 19:49, 4 July 2013
Problem
A container in the shape of a right circular cone is inches tall and its base has a
-inch radius. The liquid that is sealed inside is
inches deep when the cone is held with its point down and its base horizontal. When the liquid is held with its point up and its base horizontal, the height of the liquid is
from the base where
and
are positive integers and
is not divisible by the cube of any prime number. Find
.
Solution
Solution 1
The scale factor is uniform in all dimensions, so the volume of the liquid is of the container. The remaining section of the volume is
of the volume, and therefore
of the height when the vertex is at the top.
So, the liquid occupies of the height, or
. Thus
.
Solution 2
(Computational) The volume of a cone can be found by . In the second container, if we let
represent the height, radius (respectively) of the air (so
is the height of the liquid), then the volume of the liquid can be found by
.
By similar triangles, we find that the dimensions of the liquid in the first cone to the entire cone is , and that
; equating,
Thus the answer is , and
.
See also
2000 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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