If an error of $1^{\circ}$ is made in measuring the angle of a sector of radius $30 \mathrm{~cm}$, then the approximate error in its area, is
Answer & explanation
Correct answer: option 3
The correct answer is option 3: $2.5 \pi \mathrm{cm}^2$
Let A be the area and $\theta$ (in radians) be the sector angle. Then,
$A=\frac{1}{2} \times 30^2 \times \theta=450 ~\theta$ $\left[∵ A=\frac{1}{2} r^2 \theta\right]$
$\Rightarrow \frac{d A}{d \theta}=450$
Let $\Delta \theta$ be an error in $\theta$ and $\Delta A$ be the corresponding error in $A$. Then,
$\Delta A=\frac{d A}{d \theta} \Delta \theta$
$\Rightarrow \Delta A=450 \times \frac{\pi}{180}$ $\left[∵ \Delta \theta=1^{\circ}=\frac{\pi}{180}\right.$ radians $]$
$\Rightarrow \Delta A=2.5 \pi \mathrm{cm}^2$