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Mathematics

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 sphere formed is 6 x 6 x 6 = 216.

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

  1. Both the statement are true.

  2. Both the statement are false.

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

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

Mensuration

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Answer

Given, radius of solid sphere, R = 6 cm

Diameter of new sphere, d = 2 cm

Radius of new sphere, r = 22\dfrac{2}{2} cm = 1 cm

Volume of the sphere: 43\dfrac{4}{3} πr3

Given,

A solid sphere of radius 6 cm is melted and recast into solid spheres of diameter 2 cm each. Let no. of spheres formed be n.

∴ Volume of sphere = n × Volume of sphere recasted.

⇒ n = Volume of sphere meltedVolume of each new sphere formed\dfrac{\text{Volume of sphere melted}}{\text{Volume of each new sphere formed}}

∴ Statement 2 is true.

⇒n=43πR343πr3⇒n=R3r3⇒n=6313⇒n=2161⇒n=216.\Rightarrow n = \dfrac{\dfrac{4}{3}πR^3}{\dfrac{4}{3}πr^3}\\[1em] \Rightarrow n = \dfrac{R^3}{r^3}\\[1em] \Rightarrow n = \dfrac{6^3}{1^3}\\[1em] \Rightarrow n = \dfrac{216}{1}\\[1em] \Rightarrow n = 216.

∴ Statement 1 is true.

Thus, both the statements are true.

Hence, option 1 is the correct option.

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