Author: h.g.muller
Date: 09:31:06 02/10/06
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On February 10, 2006 at 12:14:20, Robert Hyatt wrote: >I'm not 100% certain that the >largest displacement would be produced by a sphere _totally_ contained in the >cone, although intuition says that is correct. Actually, it is not, as you can easily see from the limiting case of a very blunt cone. The hight might be 1 cm and the opening radius 1m, and a sphere that entirely immersed (radius ~0.5 cm) would hardly displace any water. A sphere double the size woud be ~half immersed, but have 8 times the volume and thus displace 4 times as much water. But a 10m-radius sphere would even do better. By a similar argument you can also see that touching at the opening rim of the cone is not generally optimal: for a very acute cone that has only ~half the sphere immersed, while a slightly smaller sphere might sink deep into the cone and be entirely immersed. The general solution has the two touching somewhere inside the cone, and the exact point is very hard to find without doing the math.
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