Question

A solid metallic sphere of diameter 21 cm is melted and recast into a number of smaller cones, each of diameter 3.5 cm and height 3 cm. Find the number of cones so formed.

Answer 1

Diameter of sphere = 21 cm
Radius of sphere $=\frac{21}{2}cm$

Volume of sphere$=\frac{4}{3}{\mathrm{\pi r}}^3=\frac{4\times 21\times 21\times 21\mathrm{\pi }}{3\times 2\times 2\times 2}=\frac{21\times 21\times 21\mathrm{\pi }}{3\times 2}{\mathrm{cm}}^3$

Diameter of the cone = 3.5 cm
Radius of the cone $=\frac{3.5}{2}=\frac{7}{4}cm$
Height = 3 cm

Volume of each cone$=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}=\frac{1}{3}\mathrm{\pi }\times 3\times {\left(\frac{7}{4}\right)}^2={\left(\frac{7}{4}\right)}^2\mathrm{\pi }{\mathrm{cm}}^3$

Total number of cones$=\frac{\mathrm{Volume}\mathrm{of}\mathrm{sphere}}{\mathrm{Volume}\mathrm{of}\text{a}\mathrm{cone}}=\frac{{\displaystyle \frac{21\times 21\times 21\mathrm{\pi }}{3\times 2}}}{{\left({\displaystyle \frac{7}{4}}\right)}^2\mathrm{\pi }}=\frac{21\times 21\times 21\times \mathrm{\pi }\times 4\times 4}{3\times 2\times \mathrm{\pi }\times 7\times 7}=504$