Question

A solid metallic sphere of radius 10.5 cm is melted and recast into a number of smaller cones, each of radius 3.5 cm and height 3 cm. Find the number of cones so formed.

Answer 1

Radius of the solid metallic sphere, r = 10.5 cm
Radius of the cone, R = 3.5 cm
Height of the cone, H = 3 cm
Let the number of smaller cones formed be n.
Volume of the metallic sphere, $V_s=\frac{4}{3}\mathrm{\pi }{\left(\mathrm{r}\right)}^3=\frac{4}{3}\mathrm{\pi }{\left(10.5\right)}^3$
Volume of the cone, $V_c=\frac{1}{3}\mathrm{\pi }{\left(\mathrm{R}\right)}^2\mathrm{H}=\frac{1}{3}\mathrm{\pi }{\left(3.5\right)}^2\times 3$
Let the number of cones thus formed be n.
$$\begin{gathered} n\times volumeofthecone=volumeofthesphere \\ \Rightarrow \frac{\mathrm{volume}\mathrm{of}\mathrm{the}\mathrm{sphere}\left({\mathrm{V}}_{\mathrm{s}}\right)}{\mathrm{volume}\mathrm{of}\mathrm{the}\mathrm{cone}\left({\mathrm{V}}_{\mathrm{c}}\right)}=n \\ \Rightarrow \frac{{\displaystyle \frac{4}{3}}\mathrm{\pi }{\left(10.5\right)}^3}{{\displaystyle \frac{1}{3}}\mathrm{\pi }{\left(3.5\right)}^2\times 3}=n \\ \Rightarrow 126=n \end{gathered}$$
Hence, 126 cones are thus formed.