A solid metallic sphere of diameter 21 cm is melted and recast into a number of smaller cones,

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.

Solution:

Diameter of sphere $=21 \mathrm{~cm}$

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

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

Diameter of the cone $=3.5 \mathrm{~cm}$

Radius of the cone $=\frac{3.5}{2}=\frac{7}{4} \mathrm{~cm}$

Height $=3 \mathrm{~cm}$

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

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