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The picture above shows a sphere sitting on a cone. The cone happens to have radius 0.7, height 2 and the sphere has radius 0.68. This picture leads one to ask the following question. Suppose a cone of given radius and height is filled with water. If a ball (sphere) is placed on top the cone, its fairly obvious that the water will overflow, and thus a certain amount of water will be displaced.by the sphere. QUESTION: What is the radius of the sphere which will maximize the volume of water displaced?

NOTES:

[a] For definiteness, you may assume the cone has radius 1 and height 3.

[b] To get started, explain in plain language why either a very large sphere or a very small sphere results in a small volume of water being displaced."

Me and my group are aboslutely stumped by this problem, any help would be appreciated.