Ex.13.3 Q1 Surface Areas and Volumes Solution - NCERT Maths Class 10

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Question

A metallic sphere of radius \(4.2\rm \,{cm}\) is melted and recast into the shape of a cylinder of radius \(6\rm \,{cm}\). Find the height of the cylinder.

Text Solution

 

What is known?

Radius of the metallic sphere \(4.2 \,\rm{cm}\) and radius of the cylinder \(6 \,\rm{cm}\)

What is unknown?

The height of the cylinder.

Reasoning:

Draw a figure to visualize the shapes better

Since, a metallic sphere is melted and recast into the shape of a cylinder then their volume must be same.

Volume of the sphere \(=\) Volume of the cylinder

 We will find the volume of the sphere and cylinder by using formulae;

Volume of the sphere\( \begin{align} = \frac{4}{3}\pi {r^3}\end{align} \)

where \(r\) is the radius of the sphere

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

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

Steps:

Radius of the hemisphere, \({r_1} = 4.2cm\)

Radius of the cylinder, \({r_2} = 6cm\)

Let the height of the cylinder be \(h\).

Volume of sphere \(=\) Volume of cylinder

\[\begin{align}\frac{4}{3}\pi r_1^3 &= \,\pi r_2^2h\\\frac{4}{3}r_1^3 &= \,r_2^2h\end{align}\]

\[\begin{align}h &= \frac{{4r_1^3}}{{3\,r_2^2}}\\&= \frac{{4 \times 4.2cm \times 4.2cm \times 4.2cm}}{{3 \times 6cm \times 6cm}}\\&= 2.74\,cm\end{align}\]

Hence, the height of the cylinder so formed will be \(2.74\rm{ cm.}\)