Ex.13.2 Q6 Surface Areas and Volumes Solution - NCERT Maths Class 10

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Question

A solid iron pole consists of a cylinder of height \(220\,\rm{cm}\) and base diameter \(24\,\rm{cm},\) which is surmounted by another cylinder of height \(60\,\rm{cm}\) and radius \(8\,\rm{cm}\).

Find the mass of the pole, given that \(1\,\rm {cm^3}\) of iron has approximately \(8\,\rm{g}\) mass.

\(\left( {{\text{Use}}\,\,\pi \,{\text{ = }}\,{\text{3}}{\text{.14}}} \right)\)

Text Solution

What is known?

A solid iron pole consisting of a cylinder of height \( 220\,\rm{cm}\) and base diameter  \(24\,\rm {cm}\)  which is surmounted by another cylinder of height  \( 60\,\rm {cm}\) and radius  \(8\,\rm {cm}\)

 Mass of \(1\;\rm{cm}^3\) iron \(= 8\rm\,{g}\) 

What is unknown?

The mass of the solid iron pole

Reasoning:

Draw the figure to visualize the iron pole

Visually it’s clear that

Volume of the solid iron pole \(=\) volume of larger cylinder \(+\) volume of smaller cylinder

Mass of iron in the pole  \(=8\,\rm{g}\) \(\;\times\;\)volume of the solid iron pole in \({\text{c}}{{\text{m}}^3}\)

We will find the volume of the solid by using formula;

Volume of the cylinder\( = \pi {r^2}h\)  

where \(r\) and \(h\) are the radius and height of the cylinder respectively.

Steps:

Radius of larger cylinder,\(\begin{align}R = \frac{{24cm}}{2} = 12cm\end{align}\)

Height of larger cylinder,\(H = 220cm\)

 Radius of smaller cylinder,\(r = 8cm\)

Height of smaller cylinder,\(h = 60cm\)

Volume of the solid iron pole \(=\) volume of larger cylinder \(+\) volume of smaller cylinder

\[\begin{align}&= \pi {R^2}H + \pi {r^2}h\\&= \pi \left( {12cm \times 12cm \times 220cm + 8cm \times 8cm \times 60cm} \right)\\&= 3.14 \times \left( {31680c{m^3} + 3840c{m^3}} \right)\\&= 3.14 \times 35520c{m^3}\\&= 111532.8c{m^3}\end{align}\]

Mass of \(1cm^3\) iron is \(8g\)

Mass of iron in the pole \(= 8g ×\) volume of the solid iron pole in \(cm^3\)

\[\begin{align}&= 8g \times 111532.8\\&= 892262.4g\\&= \frac{{892262.4}}{{1000}}kg\\&= 892.2624kg\end{align}\]

Mass of iron in the pole is \(892.26\,\rm{ kg}\)