# A solid metallic sphere of diameter 28 cm

Question:

A solid metallic sphere of diameter $28 \mathrm{~cm}$ is melted and recast into a number of smaller cones, each of diameter $4 \frac{2}{3} \mathrm{~cm}$ and height $3 \mathrm{~cm}$. Find the number of cones so formed.

Solution:

The radius of solid metallic sphere, $R=\frac{28}{2}=14 \mathrm{~cm}$

The volume of sphere

$=\frac{4}{3} \pi R^{3}$

$=\frac{4}{3} \times \pi \times(14)^{3}$

$=\frac{4}{3} \pi \times 14 \times 14 \times 14$

$=\frac{10976 \pi}{3} \mathrm{~cm}^{3}$

Given, the sphere is recast into smaller cones.

$r=\frac{14}{3 \times 2}$

$=\frac{7}{3} \mathrm{~cm}$

The height of cone h = 3 cm

Let n be the no. of smaller cones.

Clearly, the volume of solid sphere = n × volume of one smaller cone$\frac{10976}{3} \pi=n \times \frac{1}{3} \pi \times\left(\frac{7}{3}\right)^{2} \times 3$

$n \times \frac{49}{3}=10976$

$n=\frac{10976 \times 3}{49}$

$n=672$

Thus, the no. of smaller cones = 672