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### 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
```
Associated Topics:
College Higher-Dimensional Geometry
High School Higher-Dimensional Geometry

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