Volume of Cylinder

Volume of Cylinder
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As in the case of cuboids, the volume of a cylinder will depend on two factors:

  • The area A of the base

  • The height \(h\)

Thus, the volume will be

\[V = A \times h = \pi {r^2}h\]

Note that the volume of the cylinder scales linearly with the area of the base, and with the height of the cylinder. However, it scales as the square of the radius of the base. Thus, if the radius of the base is doubled, the volume will increase by a factor of 4.

Example 1: A cylinder has a radius of 3 cm and a height of 10 cm. Find its total surface area and its volume.

Solution: We have:

\[\begin{align}&{\rm{Total}}\,{\rm{SA}} = 2\pi r\left( {r + h} \right) = 78\pi \,{\rm{c}}{{\rm{m}}^2}\\ &\qquad\qquad\quad\quad\quad\; \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \approx 245\,{\rm{c}}{{\rm{m}}^2}\\&{\rm{Volume}} = \pi {r^2}h = 90\pi \,{\rm{c}}{{\rm{m}}^3}\\&\qquad\quad\quad\quad\quad\;\;\; \approx 283\,{\rm{c}}{{\rm{m}}^3}\end{align}\]

Example 2: The radius of a cylinder is 7 cm, while its volume is 1.54 L. What is the height of the cylinder? Take \(\pi  \approx \frac{{22}}{7}\).

Solution: If the height is \(h\) cm, we have:

\[\begin{align}&V = \pi {r^2}h\\ &\Rightarrow \,\,\,1540\,{\rm{c}}{{\rm{m}}^3} = \left( {\frac{{22}}{7} \times 49 \times h} \right)\,{\rm{c}}{{\rm{m}}^3}\\ &\Rightarrow \,\,\,h = 10\end{align}\]

The height of the cylinder is 10 cm.

Example 3: A cylindrical container with no lid has inner radius 20 cm and depth 10 cm. It needs to be coated on the inner walls with a paint which costs INR 6000/m2 of area. Find the cost of this paint job.

Solution: Note that the surface area S of the container which needs to be coated consists of the base of the container and the curved walls:

\[\begin{align}&S = 2\pi rh + \pi {r^2}\\&\,\,\,\, = 800\pi  \approx 2513.3\,\,{\rm{c}}{{\rm{m}}^2}\end{align}\]

The cost C of the paint job is

\[\begin{align}&C = 2513.3\,\,{\rm{c}}{{\rm{m}}^2} \times {\rm{INR}}\,0.6/{\rm{c}}{{\rm{m}}^2}\\&\,\,\,\,\, \approx {\rm{INR}}\,1508\end{align}\]

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