Hollow Circular Cylinder
Some examples of a hollow cylinder are pipes, circular buildings and bearing bushes. 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) $$V = \frac{\pi }{4}\left( {{D^2} – {d^2}} \right)$$
with $$D$$ and $$d$$ being the outer and inner diameter, and $$V$$ is volume.
Example:
A well with a 10m inside diameter is dug 14m deep. The earth taken out of it is spread all around to a width of 5m to form an embankment. Find the height of the embankment.
Solution:
Volume of the dug out earth $$ = \pi {r^2}h$$
$$ = \frac{{22}}{7} \times 5 \times 5 \times 14$$
$$1100\,{\text{cu}}{\text{.}}\,{\text{m}}$$
Area of the embankment (shaded)
$$ = \pi \left( {{R^2} – {r^2}} \right)$$
$$ = \pi \left( {{{10}^2} – {5^2}} \right)$$ $$ = 75 \times \frac{{22}}{7}\,{\text{sq}}{\text{.m}}$$
$$\therefore $$ 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 the copper used in making the pipe.
Solution:
Given that:
The height of the cylindrical pipe is $$h = 21{\text{dm}}\,\,\, = \,210{\text{cm}}$$
$$\therefore $$ 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
$$ = {\text{ Volume of external cylinder }} – {\text{ volume of internal cylinder}}$$
$$ = \pi {R^2}h – \pi {r^2}h\,\,\,\,\,\, = \,\pi \left( {{R^2} – {r^2}} \right)h\,\,\,\, = \frac{{22}}{7}\left[ {{5^2} – {3^2}} \right] \times 210$$
$$ = \frac{{22}}{7} \times 16 \times 210\,\,\,\, = 22 \times 16 \times 30\,\,\,\, = 10560\,{\text{cu}}{\text{.cm}}$$