Question

A solid metallic hemisphere of radius $8cm$ is melted and recasted into a right circular cone of base radius $6cm.$ Determine the height of the cone.

Answer 1

Radius of hemisphere $\left(r\right)=8cm$

Base radius of cone $=6cm$

The metallic hemisphere is casted into a cone

We have to determine the height of the cone

According to the given data

The volume remains same, when a object is moulded to another body

Volume of hemisphere $=$ Volume of cone

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

where$,r=$radius of hemisphere

Volume of given hemisphere $=\frac{2}{3}\times \mathrm{\pi }\times 8^3=\frac{1024}{3}\mathrm{\pi }{\mathrm{cm}}^3$

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

where$,$

$r$ is base radius

$h$ is the height of the cone.

Volume of cone $=\frac{1}{3}\mathrm{\pi }{6}^2\mathrm{h}=12\mathrm{\pi h}$

Volume of hemisphere $=$ Volume of cone

$$\begin{gathered} 12\mathrm{\pi rh}=\frac{1024\mathrm{\pi }}{3} \\ \Rightarrow 12\mathrm{h}=341.333 \\ \Rightarrow \mathrm{h}=\frac{341.333}{12} \\ \Rightarrow \mathrm{h}=28.44\mathrm{cm} \end{gathered}$$