Question

The volume of the given solid is:

 

Options

• \[\frac{4}{3} \pi r^{3}\]

• \[\frac{4}{3} \pi r^{2}\]

• \[2 \pi r^{3}\]

• \[4 \pi r^{3}\]

Answer 1

\[\boldsymbol{2\pi r^{3}}\]

Explanation:

From figure,

Height of cone (h) = r

Height of cylinder (H) = r

Radius of hemispherical portion = Radius of cylindrical portion = Radius of conical portion = r.

From figure,

Volume of body = Volume of conical portion + Volume of cylindrical portion + Volume of hemispherical portion

\[{} = \frac{1}{3}\, \pi r^2 h + \pi r^2 H + \frac{2}{3}\, \pi r^3\]

\[{} = \frac{1}{3}\, \pi r^2 \cdot r + \pi r^2 \cdot r + \frac{2}{3}\, \pi r^3\]

\[{} = \frac{1}{3}\, \pi r^3 + \pi r^3 + \frac{2}{3}\, \pi r^3\]

\[{} = \frac{\pi r^3 + 3\pi r^3 + 2\pi r^3}{3}\]

\[{} = \frac{6\pi r^3}{3}\]

= 2πr³