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

Solution:

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

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

$x=28 \mathrm{~cm}$

$\Delta x=0.01 \mathrm{~cm}$

$x=2 \pi r$

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

$\Rightarrow \frac{d y}{d x}=\frac{x}{2 \pi}$

$\Rightarrow \frac{\Delta y}{y}=\frac{x}{2 \pi y} d x=\frac{2}{x} \times 0.01$

$\Rightarrow \frac{\Delta y}{y} \times 100=\frac{2}{x}=\frac{1}{14}$

Hence, the percentage error in the area is $\frac{1}{14}$.