Question

A solid sphere of radius 6 cm is melted and recast into solid spheres of diameter 2 cm each.

Statement (1): The number of solid spheres formed is 6 × 6 × 6 = 216.

Statement (2): If the smaller spheres formed are identical, the number of solid spheres formed \[=\frac{\text { Volume of sphere melted }}{\text { Volume of each of the sphere formed }}\]

विकल्प

• Both the statements are true.

• Both the statements are false.

• Statement 1 is true, and statement 2 is false.

• Statement 1 is false, and statement 2 is true.

Answer 1

Both the statements are true.

Explanation:

Given, radius of solid sphere, R = 6 cm

Diameter of new sphere, d = 2 cm

Radius of new sphere, \[r = \frac{2}{2}\ \text{cm} = 1\ \text{cm}\]

Volume of the sphere: \[\frac{4}{3}\, \pi r^3\]

Given,

A solid sphere of radius \[6\ \text{cm}\] is melted and recast into solid spheres of diameter \[2\ \text{cm}\] each. Let no. of spheres formed be \[n.\]

∴ Volume of sphere = \[n\] × Volume of sphere recasted.

\[\Rightarrow n = \frac{\text{Volume of sphere melted}}{\text{Volume of each new sphere formed}}\]

∴ Statement 2 is true.

\[\Rightarrow n = \frac{\frac{4}{3}\, \pi R^3}{\frac{4}{3}\, \pi r^3}\]

\[\Rightarrow n = \frac{R^3}{r^3}\]

\[\Rightarrow n = \frac{6^3}{1^3}\]

\[\Rightarrow n = \frac{216}{1}\]

\[\Rightarrow n = 216.\]

∴ Statement 1 is true.

Thus, both the statements are true.