Author: Mridul Muralidharan
Date: 16:19:04 02/10/06
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On February 10, 2006 at 12:31:06, h.g.muller wrote: >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. The sphere is dropped 'into the cone' - so submerged , not floating. Hence solution is to find the incentre of the triangle (2d-ification of the cone) and use that to find the radius. Mridul
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