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
(i) radius $r^{\prime }$ of the new sphere, (ii) ratio of $S$ and $S^{\prime }$

Answer 1

Assume that matter is conserved in the process. So, the total volume of 27 spheres is equal to the volume of the new sphere.

(i)

$27\frac{4\pi r^3}{3}=\frac{4\pi r^{\prime 3}}{3}$

$\therefore r^{\prime }=3r$

(ii)

We have

$S=4\pi r^2$

$S^{\prime }=4\pi r^{\prime 2}$

$\frac{S}{S^{\prime }}=\frac{4\pi r^2}{4\pi r^{\prime 2}}$

$=(\frac{r}{r^{\prime }}{)}^2$

$=(\frac{r}{3r}{)}^2$

$=\frac{1}{9}$