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

 

Options

• \[\frac{1}{14}\]

• 0.01

• \[\frac{1}{7}\]

• none of these

Answer 1

\[\frac{1}{14}\]

Let x be the radius of the circle and y be its circumference.

\[x = 28 cm\]

\[ ∆ x = 0 . 01 cm\]

\[x = 2\pi r\]

\[y = \pi r^2 = \pi \times \frac{x^2}{4 \pi^2} = \frac{x^2}{4\pi}\]

\[ \Rightarrow \frac{dy}{dx} = \frac{x}{2\pi}\]

\[ \Rightarrow \frac{∆ y}{y} = \frac{x}{2\pi y}dx = \frac{2}{x} \times 0 . 01\]

\[ \Rightarrow \frac{∆ y}{y} \times 100 = \frac{2}{x} = \frac{1}{14}\]

\[\text { Hence, the percentage error in the area is } \frac{1}{14} .\]