Author: Randall Shane
Date: 17:50:22 02/10/06
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On February 10, 2006 at 19:19:04, Mridul Muralidharan wrote: >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 The problem is not to find the largest spehere that is totally submerged, it to find the sphere that displaces the largest amount of water when it sinks.
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