Problem 29 · 2023 Math Kangaroo
Stretch
Geometry & Measurement
areaarea-decomposition
Two identical cylindrical glasses contain the same amount of water. The left glass is upright, while the right one rests against the other one at a slant. The water level in both glasses is at the same height. The water level in the leaning glass touches its bottom in exactly one point (see diagram). The bases of both glasses have an area of \(3\pi\) cm². How much water is in each glass?
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Answer: A — \(9\pi\) cm³
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Hint 1 of 2
The leaning glass holds a wedge of water whose surface passes through the single bottom contact point.
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Hint 2 of 2
Compare that wedge to the upright cylinder of the same equal water height.
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Approach: equate the wedge volume to the upright cylinder volume
- The base area is 3π, so the radius squared is 3. The water heights are equal in both glasses.
- The wedge in the tilted glass, with its surface through the lone bottom point, has the same volume as a cylinder of that height on the 3π base.
- Working it through, each glass holds 9π cm3.
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