Question

Twenty-seven solid iron spheres, each of radius $r$ and surface area $S$ are melted to form a sphere with surface area $S\text{'}$. Find the radius $r\text{'}$ of the new sphere,

Answer 1

Let's find the radius of the new sphere.

The volume of the solid sphere $=\frac{4}{3}{\mathrm{\pi r}}^3$

Hence, the volume of twenty-seven solid sphere $=27\times \frac{4}{3}{\mathrm{\pi r}}^3=36{\mathrm{\pi r}}^3$

New solid iron sphere radius $=r\text{'}$

By using the formula, the volume of this new sphere $=\frac{4}{3}{\mathrm{\pi r}}^{\text{'}3}$

$\begin{array}{rcl}\frac{4}{3}{\mathrm{\pi r}}^{\text{'}3}& =& 36{\mathrm{\pi r}}^3\\ & \Rightarrow & r^{\text{'}3}=27{\mathrm{\pi r}}^3\\ & \Rightarrow & {\mathrm{r}}^{\text{'}3}=3\mathrm{r}\\ & & \end{array}$

Therefore, the radius of new sphere $=3r$