Question

If the relative error in measuring the radius of a circular plane is α, find the relative error in measuring its area.

Answer 1

Let x be the radius and y be the area of the circular plane.
$$\begin{gathered} \mathrm{We}\mathrm{have}\frac{△x}{x}=\alpha \mathrm{and}y=x^2. \\ \Rightarrow \frac{dy}{dx}=2x \\ \Rightarrow \frac{△y}{y}=\frac{2x}{y}dx=\frac{2x}{x^2}\times \alpha x=2\alpha \\ \mathrm{Hence},\mathrm{the}\mathrm{relative}\mathrm{error}\mathrm{in}\mathrm{the}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{circular}\mathrm{plane}\mathrm{is}2\mathrm{\alpha }. \end{gathered}$$