HMMT 2002 U30
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A conical flask contains some water. When the flask is oriented so that its base is horizontal and lies at the bottom (so that the vertex is at the top), the water is 1 inch deep. When the flask is turned upside-down, so that the vertex is at the bottom, the water is 2 inches deep. What is the height of the cone?
Dataset text NuminaMath olympiads. Quoted with source.
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\frac{1}{2}+\frac{\sqrt{93}}{6}
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