Question

The surface area of a solid metallic sphere is $616$ $c{m}^2$. It is melted and recast into smaller spheres of diameter $3.5$ cm. How many such spheres can be obtained?

Answer 1

Given:

Surface area of bigger sphere $=616c{m}^2$

$\Rightarrow 4\pi R^2=616$

$\Rightarrow 4\times \frac{22}{7}\times R^2=616$

$\Rightarrow R^2=\frac{616}{4}\times \frac{7}{22}$

$\Rightarrow R=7cm$

Volume of bigger sphere $=\frac{4}{3}\pi R^3$
$=\frac{4}{3}\times \frac{22}{7}\times 7\times 7\times 7$ $...\left(1\right)$

Diameter of small sphere $=3.5cm$ [Given]

Radius of small sphere $\left(r\right)=\frac{3.5}{2}$

Volume of small sphere $=\frac{4}{3}\pi r^3$

$=\frac{4}{3}\times \frac{22}{7}\times \frac{3.5}{2}\times \frac{3.5}{2}\times \frac{3.5}{2}$ $...\left(2\right)$

Then the number of small spheres $=\frac{Volumeofbigsphere}{Volumeofsmallspheres}$

$=\frac{\frac{4}{3}\times \frac{22}{7}\times 7\times 7\times 7}{\frac{4}{3}\times \frac{22}{7}\times \frac{3.5}{2}\times \frac{3.5}{2}\times \frac{3.5}{2}}$

$=64$