# 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) $$ \Rightarrow \frac{4}{3}\pi {r^3} = 972\pi $$

$$ \Rightarrow {r^3} = 729 \Rightarrow r = 9$$