What Fraction of Water Overflows?

Date: 02/04/2002 at 03:54:09
From: Asim Shaukat
Subject: Mensuration

A conical vessel of radius 6cm and height 8cm is completely filled 
with water. A sphere is lowered into it and its size is such that when 
it touches the sides of the conical vessel it is JUST immersed. What 
fraction of water overflows?

I got the correct answer (which is 37.5%) by assuming the radius of 
the sphere to be half that of the conical vessel. However, I am not 
able to DERIVE the solution by applying any theories.

Date: 02/04/2002 at 16:39:37
From: Doctor Peterson
Subject: Re: Mensuration

Hi, Asim.

A cross section will show a circle of radius r inscribed in a 
triangle, with R=6 and H=8:

                                    R
    +----------------oooo+oooo----------------+
     \            ooo    |    ooo            /
      \         oo       |       oo         /
       \       o         |         o       /
        \     o         r|          o     /
         \   o           |           o   /
          \ o            |            o /
           \o            +            o/
            \            |  \  r      /
             \           |      \    /
              \          |          +
               \         |         /
                \o    H-r|       o/
                 \ooo    |    ooo/
                  \  oooo|oooo  /
                   \     |     /
                    \    |    /
                     \   |   /
                      \  |  /
                       \ | /
                        \|/
                         +

Look for a pair of similar triangles, and use them to write an 
equation that you can solve for r, which for the particular numbers 
you have will indeed be R/2, though that would not be true in general. 
Then find the ratio of the volume of the sphere to the volume of the 
cone, as you evidently did already.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 02/04/2002 at 22:39:11
From: Asim Shaukat
Subject: Mensuration

Dr. Peterson,

I appreciate the help very much.
Thanks and regards,
Asim