Question

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

Answer 1

$r=1cm$

$h=1cm$

We have to find out the volume of the solid in terms of $\mathrm{\pi }$

Total Volume$=$ Volume of cone $+$ Volume of Hemisphere

We know that:

Volume of cone$\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\mathit{\pi }{\mathit{r}}^{\mathbf{2}}\mathbf{h}$

Volume of hemisphere $=\frac{2}{3}{\mathrm{\pi r}}^3$

$$\begin{gathered} =\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}+\frac{2}{3}{\mathrm{\pi r}}^3 \\ =\frac{1}{3}{\mathrm{\pi r}}^2(\mathrm{h}+2\mathrm{r}) \\ =\frac{1}{3}\times \mathrm{\pi }\times 1\times 1\times (1+2) \\ =\mathrm{\pi }{\mathrm{cm}}^3 \end{gathered}$$

Hence, the volume of the solid is $\mathrm{\pi }{\mathrm{cm}}^3$.