A spherical ball in a conical wine glass

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Question from Jules, a student:

Dear QQ,
I have the following question :

" A heavy spherical ball is lowered carefully into a full conical wine
glass whose depth is h and whose generating angle (between the axis
and a generator) is w. Show that the greatest overflow occurs when the
radius of the ball is (h*sin(w))/(sin(w)+cos(2w))."

Thanks!

I have solved this problem by taking a cross section when the ball is
fully immersed and by using 2 similar triangles and the pythagorean theorem

I get radius of ball = (h*sin(w))/(1+sin(w))

which is not the same answer as given in the problem. Why ?
Thanks for your help!

Jules,

It is not clear to me that the ball should be be fully immersed for the largest overflow to occur. If you have h small and w very wide, this gives a very small ball, while a bigger mess would be made by a big ball which touches the rim, and is tangent to the side at the rim.

Claude

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