Question

If the percentage error in the radius of a sphere is $\alpha$, find the percentage error in its volume.

Answer 1

Given : $\frac{\Delta r}{r}\times 100=\alpha$

$\Rightarrow \Delta r=0.01\alpha r$

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

$\Rightarrow \frac{dV}{dr}=\frac{d}{dr}\left(\frac{4}{3}\pi r^3\right)=\frac{4}{3}\pi .3r^2$

$\Rightarrow \frac{dV}{dr}=4\pi r^2$

$\because \Delta V\approx \frac{dV}{dr}\times \Delta r$

$\Rightarrow \Delta V\approx 4\pi r^2\times 0.01\alpha r$

$\Rightarrow \Delta V\approx 0.04\alpha \pi r^3$

Now, percentage error in volume is :

$\frac{\Delta V}{V}\times 100\%$

$\approx \frac{0.04\alpha \pi r^3}{\frac{4}{3}\pi r^3}\times 100\%$

$\approx 3\alpha \%$

$\therefore$ Percentage error in volume of sphere is $3\alpha \%$.
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