Ex.13.2 Q1 Surface Areas and Volumes Solution - NCERT Maths Class 10

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Question

A solid is in the shape of a cone standing on a hemisphere with both their radii Being equal to \(1\,\rm{cm}\) and the height of the cone is equal to its radius. Find the volume of the solid in terms of \(\pi \).

Text Solution

 

What is known?

A solid in the shape of a cone standing on a hemisphere. Radius and height of the cone are same as the radius of the hemisphere which is \(1\rm{cm}\).

What is unknown?

The volume of the solid.

Reasoning:

 Draw a figure of the solid to visualize it

Since the solid is made up of conical part and a hemispherical part.

Volume of the solid \(=\) volume of the conical part \(+\) volume of the hemispherical part

We will find the volume of the solid by using formulae;
Volume of the hemisphere  \( \begin{align} = \frac{2}{3}\pi {r^3}\end{align} \)
where \(r\) is the radius of the hemisphere
Volume of the cone   \( \begin{align} = \frac{1}{3}\pi {r^2}h\end{align} \)
where \(r\) and \(h\) are the radius and height of the cone respectively.

Steps:

Radius of hemispherical part \(=\) Radius of conical part \(=r=1\,\rm{cm}\)

Height of conical part \( = h = r = 1cm\)

Volume of the solid  \(=\) volume of the conical part \(+\) volume of the hemispherical part

\[\begin{align}&= \frac{1}{3}\pi {r^2}h + \frac{2}{3}\pi {r^3}\\&= \frac{1}{3}\pi {r^3} + \frac{2}{3}\pi {r^3}\\&= \pi {r^3}\\ &= \pi {\left( {1cm} \right)^3}\\&= \pi c{m^3}\end{align}\]