Hello, snigdha!
A conical vessel of radius 6 cm and height 8 cm is completely filled with water.
A sphere is lowered into the water and its size is such that when it touches the sides,
it is just immersed. .What fraction of water will overflow? Code:
A 6 P 6 B
- o - - - - - - * o * - - - - - - o
:\ * | * /
: \ * |r * /
: \ * | * /
: \ | /
: \* oO */ 6
: \ | * r /
: * | * *
8 : * |r * *
: * | o Q
: \ * o * /
: \ | /
: \ | /
: \ | / 4
: \ | /
: \|/
- o
C
is the cone, with altitude 
Its radius is: . 
In right triangle 
, Pythagorus tell us: . 
Note that: . 
. . Hence: . 
The center of the sphere is 
with radius 
In right triangle 
We have: . 
Then: . ^2 \quad\Rightarrow\quad 16r \:=\:48 \quad\Rightarrow\quad\boxed{ r \:=\:3})
The volume of the sphere is: .  \:=\:36\pi)
. . This is the volume of water that will overflow.
The volume of the cone is: . (8) \:=\:96\pi)
The fraction is: . 
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