The relative error in the measurement of the volume of a sphere is $\Delta V/V$. If the relative err

Question

The relative error in the measurement of the volume of a sphere is $\Delta V/V$. If the relative error in the measurement of its radius is $\Delta r/r$, then what is the relationship between them?

Professor PC Thomas Classes · Answer

Volume of a sphere is $V = \frac{4}{3}\pi r^3$. The relative error in a quantity raised to a power is the power times the relative error in the base quantity. Thus, $\frac{\Delta V}{V} = 3 \frac{\Delta r}{r}$.