A solid metallic sphere of radius 12 cm  is melted and recast in the form of small spheres of radius 2 cm. How many small spheres are formed?

Correct option is B

Given:

Radius of larger sphere = 12cm

Radius of smaller sphere = 2cm

Formula Used:

The volume V of a sphere is given by the formula:

​$$V = \frac{4}{3} \pi r^3$$ 

Where: r is radius of a sphere 

Solution: 

For the large sphere, the radius is 12 cm. So, the volume of the large sphere is:

​$$V_{\text{large}} = \frac{4}{3} \pi (12)^3$$ 

$$V_{\text{large}} = \frac{4}{3} \pi \times 1728 = \frac{6912}{3} \pi = 2304 \pi \, \text{cm}^3$$ 

​the volume of one of the small spheres, which has a radius of 2 cm. Using the same formula for the volume of a sphere:

​$$V_{\text{small}} = \frac{4}{3} \pi (2)^3$$ 

$$V_{\text{small}} = \frac{4}{3} \pi \times 8 = \frac{32}{3} \pi \, \text{cm}^3$$ 

​Let $N$ be the number of small spheres. Then:

$$N \times V_{\text{small}} = V_{\text{large}}$$ 

​$$N \times \frac{32}{3} \pi = 2304 \pi$$​​

​$$N \times \frac{32}{3} = 2304$$​​

$$N \times 32 = 6902$$​ 

$$N = \frac{6902}{32} = 216$$​ 

Thus the number of small spheres formed is 216.