Problem:
An inverted cone with base radius and height is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of . What is the height in centimeters of the water in the cylinder?
Answer Choices:
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B.
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E.
Solution:
The volume of a cone is where is the base radius and is the height. The water completely fills up the cone so the volume of the water is .
The volume of a cylinder is so the volume of the water in the cylinder would be .
We can equate these two expressions because the water volume stays the same like this . We get and .
So the answer is .
The water completely fills up the cone. For now, assume the radius of both cone and cylinder are the same. Then the cone has of the volume of the cylinder, and so the height is divided by . Then, from the problem statement, the radius is doubled, meaning the area of the base is quadrupled (since ).
Therefore, the height is divided by and divided by , which is .
The problems on this page are the property of the MAA's American Mathematics Competitions