Question

Twenty seven solid iron spheres , each of radius $r$ and surface area $S$ are melted to form a sphere with surface area $S^{{}^{\prime }}$ find the

1) radius $r^{{}^{\prime }}$ of the new sphere
2) ratio of $S$ and $S^{{}^{\prime }}$

Answer 1

Volume of 27 spheres = volume of 1 big sphere

$\Rightarrow \frac{4}{3}\pi (r{)}^3.27=\frac{4}{3}\pi (r^1{)}^3$

$\Rightarrow (3r{)}^3=(r^1{)}^3$

So, $r^1=3r$

$\frac{s}{s^1}=\frac{4\pi r^2}{4\pi (r^1{)}^2}={\left(\frac{r}{r^1}\right)}^2={\left(\frac{1}{3}\right)}^2=\frac{1}{9}$