Question

The radius of a sphere is 10 cm. If the radius is increased by 1 cm, then prove that volume of the sphere increases by 33.1$\%$.

Answer 1

Given that, a sphere has a radius of $10cm$.

This radius is increased by $1cm$.

Hence,

$r_1=10cm$

$r_2=11cm$

We know that, volume of a sphere $=\frac{4}{3}\pi r^3$.

Hence, difference in volume $=\Delta v=\frac{4}{3}\pi (r_2^3-r_1^3)$

$\therefore \%\text{increase}=\frac{\Delta v}{v}\times 100=\left(\frac{r_2^3-r_1^3}{r_1^3}\right)\times 100$

$\Rightarrow \left(\frac{{11}^3-{10}^3}{{10}^3}\right)\times 100=\left(\frac{1331-1000}{1000}\right)\times 100$

$\Rightarrow 0.331\times 100=33.1\%$.

Hence, it is proved that if the radius of the given sphere is increased by $1cm,$ its volume increases by $33.1\%$.