Question

If the radius of a sphere is increased by 10%, prove that its volume will be increased by 33.1%.

Answer 1

Suppose that r is the radius of the sphere.
Volume of the original sphere = $\frac{4}{3}\mathrm{\pi }{r}^3=V\left(\mathrm{say}\right)$
Let us increase the radius of the sphere by 10%.
Then new radius = $\left(r+\frac{10}{100}r\right)$ = $\frac{110}{100}\times r=\frac{11r}{10}$

$$\begin{gathered} \mathrm{Now},\mathrm{new}\mathrm{volume}\mathrm{of}\mathrm{the}\mathrm{sphere}=\frac{4}{3}\mathrm{\pi }{\left(\frac{11r}{10}\right)}^3 \\ =\frac{4}{3}\mathrm{\pi }\times \frac{1331}{1000}{r}^3 \\ =\frac{1331}{1000}V \\ \mathrm{Hence},\mathrm{increase}\mathrm{in}\mathrm{volume}=\frac{1331V}{1000}\mathit{-}V \\ =\frac{331V}{1000} \\ \therefore \mathrm{Increase}\%=\frac{\frac{331V}{1000}}{V}\times 100=33.1\% \end{gathered}$$

Hence, proved