If the percentage error in the radius of a sphere is α, find the percentage error in its volume.

Question:

If the percentage error in the radius of a sphere is α, find the percentage error in its volume.

Solution:

Let V be the volume of the sphere.

$V=\frac{4}{3} \pi x^{3}$

We have

$\frac{\Delta x}{x} \times 100=\alpha$

$\Rightarrow \frac{d V}{d x}=4 \pi x^{2}$

$\Rightarrow \frac{d V}{V}=\frac{4 \pi x^{2}}{V} d x$

$\Rightarrow \frac{\Delta V}{V}=\frac{4 \pi x^{2}}{\frac{4}{3} \pi x^{2}} \times \frac{x \alpha}{100}$

$\Rightarrow \frac{\Delta V}{V} \times 100=3 \alpha$

Hence, the the percentage error in the volume is $3 \alpha$.

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