Question

If r₁ and r₂ be the radii of two solid metallic spheres and if they are melted into one solid sphere, prove that the radius of the new sphere is \[\left( r_1^3 + r_2^3 \right)^\frac{1}{3}\].

Answer 1

Volume of first sphere `=4/3 pir_1^3`

Volume of second sphere `=4/3 pir_2^3`

Total volume of new sphere  `=(4/3 pir_1^3 +=4/3 pir_2^3)` 

Say of radius of new sphere = r₃

Volume of new sphere `=4/3 pir_3^3`

Hence,

`4/3 pir_3^3 =4/3 pir_1^3+4/3 pir_2^3`

`4/3 pir_3^3 =4/3 pi(r_1^3+r_2^3)`

      `r_3^3 = r_1^3 +r_2^3`

So, radius of new sphere `r_3 = (r_1^3 + r_2^3)^(1/3)`.