# Spherical Shell

Solid enclosed between two concentric spheres is called a spherical shell. For a spherical shell if $R$ and $r$ are outer and inner radii respectively, then the volume of the shell is
$= \frac{4}{3}\pi \left( {{R^3} - {r^3}} \right)$
Or        $= \frac{\pi }{6}\left( {{D^3} - {d^3}} \right)$

Example:

A sphere of radius 5cm is dropped into a cylindrical vessel partly filled with water. The diameter of the vessel is 10cm. If the sphere is completely submerged, by how much will the surface of water be raised?

Solution:

The sphere will displace a volume of water equal to that of itself and this amount of water will go up.
$\therefore$  Volume of sphere $= \frac{4}{3}\pi {r^3} = \frac{4}{3}\pi {\left( {\frac{5}{2}} \right)^3} = \frac{{125}}{6}\pi$cu.cm
The radius of the cylinder vessel $= \frac{{10}}{2} = 5$cm
The volume height occupied by $\frac{{125}}{6}\pi$cu.cm. of water in a cylindrical vessel of 5cm radius $= \frac{{125}}{6}\pi + \pi {\left( 5 \right)^3}$
$= \frac{{125}}{{6 \times 25}} = \frac{5}{6}$cm

Example:

A solid cylinder of glass, the radius of whose base is 9cm and height 12cm is melted and turned into sphere. Find the radius of the sphere so formed.

Solution:
Volume of the cylinder $= \pi {r^2}h = \pi \times {\left( 9 \right)^2} \times 12$
$= 972\pi$cu.cm       --- (1)
Volume of the sphere   $= \frac{4}{3}\pi {r^3}$                 --- (2)
By the given condition,
(1) = (2) $\Rightarrow \frac{4}{3}\pi {r^3} = 972\pi$
$\Rightarrow {r^3} = 729 \Rightarrow r = 9$