If an error of 1 ° is made in measuring the angle of a sector of radius 30 cm, then the approximate error in its area, is

A
$450 {cm}^{2}$
B
$25 π {cm}^{2}$
C
$2.5 π {cm}^{2}$
D
none of these

Solution:

Let $A$ be the area and 0 (in radians) be the sector angle. Then,

$\begin{array}{l} A = \frac{1}{2} \times 30^{2} \times θ = 450 θ \\ = \frac{d A}{d θ} = 450 \end{array} [∵ A = \frac{1}{2} r^{2} θ]$

Let $Δ θ$ be an error in $θ$ and $ΔA$ be the corresponding error in $A .$

Then,
$\begin{array}{l} Δ A = \frac{d A}{d θ} Δ θ \\ = Δ A = 450 \times \frac{π}{180} [∵ Δ θ = 1^{\circ} = \frac{π}{180} radians] \\ = Δ A = 2.5 π {cm}^{2} \end{array}$

Solution:

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