Author: Mridul Muralidharan
Date: 23:33:25 02/10/06
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On February 10, 2006 at 20:50:22, Randall Shane wrote: >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. Both are the same problem. The largest amount of liquid displaced == largest volume of the sphere == same as what I described. Otherwise , you will need to know details of the density of the sphere , etc to calculate the sphere which can by partly submerged and yet still displace more water.
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