Question

r₁, r₂ and r₃ are the radii of three metallic spheres. If these spheres are melted to form a single solid sphere, the radius of sphere formed is:

Options

• \[\sqrt{r_{1}^{3}+r_{2}^{3}+r_{3}^{3}}\]

• \[r_{1}+r_{2}+r_{3}\]

• \[r_{1}^{3}+r_{2}^{3}+r_{3}^{3}\]

• \[\sqrt[3]{r_{1}^{3}+r_{2}^{3}+r_{3}^{3}}\]

Answer 1

\[\boldsymbol{\sqrt[3]{r_{1}^{3} + r_{2}^{3} + r_{3}^{3}}}\]

Explanation:

Given,

r₁, r₂ and r₃ are the radii of three metallic spheres. These spheres are melted to form a single solid sphere.

Let the radius of sphere formed be R.

∴ Volume of sphere formed = Sum of volume of three smaller spheres

\[\Rightarrow \frac{4}{3}\, \pi R^3 = \frac{4}{3}\, \pi r_1^3 + \frac{4}{3}\, \pi r_2^3 + \frac{4}{3}\, \pi r_3^3\]

\[\Rightarrow \frac{4}{3}\, \pi R^3 = \frac{4}{3}\, \pi (r_1^3 + r_2^3 + r_3^3)\]

\[\Rightarrow R^3 = r_1^3 + r_2^3 + r_3^3\]

\[\Rightarrow R = \sqrt[3]{r_1^3 + r_2^3 + r_3^3}\]