2000 AIME I Problems/Problem 8
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 .
Contents
[hide]Solution
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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 | |
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