Question

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

Answer 1

Let V be the volume of the sphere.
$$\begin{gathered} V=\frac{4}{3}\pi x^3 \\ \mathrm{We}\mathrm{have} \\ \frac{∆x}{x}\times 100=\alpha \\ \Rightarrow \frac{dV}{dx}=4\pi x^2 \\ \Rightarrow \frac{dV}{V}=\frac{4\pi x^2}{V}dx \\ \Rightarrow \frac{∆V}{V}=\frac{4\pi x^2}{\frac{4}{3}\pi x^3}\times \frac{x\alpha }{100} \\ \Rightarrow \frac{∆V}{V}\times 100=3\alpha \\ \mathrm{Hence},\mathrm{the}\mathrm{the}\mathrm{percentage}error\mathrm{in}\mathrm{the}\mathrm{volume}\mathrm{is}3\mathrm{\alpha }. \end{gathered}$$