Question

The circumference of a circle is measured as 28 cm with an error of 0.01 cm. The percentage error in the area is

(a) $\frac{1}{14}$

(b) 0.01

(c) $\frac{1}{7}$

(d) none of these

Answer 1

(a) $\frac{1}{14}$
Let x be the radius of the circle and y be its circumference.
$$\begin{gathered} x=28\mathrm{cm} \\ ∆x=0.01\mathrm{cm} \\ x=2\pi r \\ y=\pi r^2=\pi \times \frac{x^2}{4{\pi }^2}=\frac{x^2}{4\pi } \\ \Rightarrow \frac{dy}{dx}=\frac{x}{2\pi } \\ \Rightarrow \frac{∆y}{y}=\frac{x}{2\pi y}dx=\frac{2}{x}\times 0.01 \\ \Rightarrow \frac{∆y}{y}\times 100=\frac{2}{x}=\frac{1}{14} \\ \mathrm{Hence},\mathrm{the}\mathrm{percentage}\mathrm{error}\mathrm{in}\mathrm{the}\mathrm{area}\mathrm{is}\frac{1}{14}. \end{gathered}$$