Filled To The Rim

Calculus Level 4

Suppose an inverted cone of radius 3 and height 4 is filled with water. (The central axis is perpendicular to the ground, allowing the water to reach the entirety of the cone's rim.) A sphere of radius $$r$$ made of a material denser than water is then placed gently in the water and allowed to come to rest in contact with the cone, displacing water as it settles. (That is, the volume of water displaced is the same as the volume of the settled sphere that lies below the plane that includes the rim of the cone.)

The radius of the sphere that results in the maximum displacement of water can be expressed as $$r= \dfrac mn$$, where $$m$$ and $$n$$ are coprime positive integers. Find $$m+n$$.

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