Question

A metallic sphere of radius 10.5 cm is melted and then recast into smaller cones, each of radius 3.5 cm and height 3 cm. How many cones are obtained?

Answer 1

Radius of the metallic sphere, r = 10.5 cm
Volume of the sphere = $\frac{4}{3}\mathrm{\pi }{r}^3\mathit{=}\frac{4}{3}\mathrm{\pi }(10.5{)}^3{\mathrm{cm}}^3$
Radius of each smaller cone, r₂ = 3.05 cm
Height of each smaller cone = 3 cm
$\mathrm{Volume}\mathrm{of}\mathrm{each}\mathrm{smaller}\mathrm{cone}=\frac{1}{3}\mathrm{\pi }{r_2}^{2}h=\frac{1}{3}\mathrm{\pi }(3.05{)}^2\times 3{\mathrm{cm}}^3$
Number of cones obtained = $\frac{\mathrm{volume}\mathrm{of}\mathrm{the}\mathrm{sphere}}{\mathrm{volume}\mathrm{of}\mathrm{each}\mathrm{smaller}\mathrm{cone}}$
$$\begin{gathered} =\frac{{\displaystyle \frac{4}{3}}\mathrm{\pi }{r}^{\mathit{3}}}{{\displaystyle \frac{1}{3}}\mathrm{\pi }{r_2}^{\mathit{2}}h} \\ =\frac{4\times 10.5\times 10.5\times 10.5}{3.5\times 3.5\times 3} \\ =126.006\approx 126 \end{gathered}$$

∴ 126 cones are obtained.