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 :

1. $\sqrt{r$1^3 + r2^3 + r_3^3}

2. r₁ + r₂ + r₃

3. $r$1^3 + r2^3 + r_3^3

4. $\sqrt[3]{r$1^3 + r2^3 + r_3^3}

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 \dfrac{4}{3}πR^3 = \dfrac{4}{3}πr$1^3 + \dfrac{4}{3}πr2^3 + \dfrac{4}{3}πr3^3 \\[1em] \Rightarrow \dfrac{4}{3}πR^3 = \dfrac{4}{3}π(r1^3 + r2^3 + r3^3) \\[1em] \Rightarrow R^3 = r1^3 + r2^3 + r3^3 \\[1em] \Rightarrow R = \sqrt[3]{r1^3 + r2^3 + r3^3}.

Hence, Option 4 is the correct option.