# Hollow Circular Cylinder

The example of hollow cylinders is pipes, circular buildings and bearing bushes, etc. If $R$ is the outside radius of the cylinder and $r$ is the inside radius of the cylinder, then

(i)         $V = \pi {R^2}h - \pi {r^2}h = \pi \left( {{R^2} - {r^2}} \right)h$
(ii)        $V = \frac{\pi }{4}\left( {{D^2} - {d^2}} \right)$

$D$ and $d$ being outer and inner diameter, $V$is volume.

Example:

A well with 10m inside diameter dug 14m deep. Earth taken out of it is spread all round to a width of 5m to form an embankment. Find the height of the embankment.

Solution:
Volume of the dug out           $= \pi {r^2}h$
$= \frac{{22}}{7} \times 5 \times 5 \times 14$

$1100\,{\text{cu}}{\text{.}}\,{\text{m}}$

$= \pi \left( {{R^2} - {r^2}} \right)$
$= \pi \left( {{{10}^2} - {5^2}} \right)$           $= 75 \times \frac{{22}}{7}\,{\text{sq}}{\text{.m}}$

$\therefore$            Height of the embankment

$= \frac{{{\text{Volume of the earth dug out}}}}{{{\text{Area of the embankment}}}}$

$= \frac{{1100}}{{75 \times \frac{{22}}{7}}}\,\, = \frac{{14}}{3}\,\,\, = 4\frac{2}{3}\,\,{\text{m}}$

Example:

A hollow cylinder copper pipe is 21dm long. Its outer and inner diameters are 10cm and 6cm respectively. Find the volume of copper used in making the pipe.

Solution:

Given that:
Height of cylindrical pipe, $h = 21{\text{dm}}\,\,\, = \,210{\text{cm}}$

$\therefore$            External radius, $R = \frac{{10}}{2}\,\, = \,5{\text{cm}}$
Internal radius, $R = \frac{6}{2}\,\, = \,3{\text{cm}}$
Volume of the copper used in making the pipe
$= {\text{ Volume of external cylinder }} - {\text{ volume of internal cylinder}}$
$= \pi {R^2}h - \pi {r^2}h\,\,\,\,\,\, = \,\pi \left( {{R^2} - {r^2}} \right)h\,\,\,\, = \frac{{22}}{7}\left[ {{5^2} - {3^2}} \right] \times 210$
$= \frac{{22}}{7} \times 16 \times 210\,\,\,\, = 22 \times 16 \times 30\,\,\,\, = 10560\,{\text{cu}}{\text{.cm}}$