# Spherical Shell

A solid enclosed between two concentric spheres is called a spherical shell. For a spherical shell, if $R$ and $r$ are the 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, how much will the water rise?

Solution:

The sphere will displace a volume of water equal to that of itself, and this shows how much the water will rise.

$\therefore$  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 cylindrical 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 is 12cm, is melted and turned into a sphere. Find the radius of the sphere which is 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) $\Rightarrow \frac{4}{3}\pi {r^3} = 972\pi$
$\Rightarrow {r^3} = 729 \Rightarrow r = 9$