[The area of a sector is $\frac{1}{2} r^2 \theta$, where $\theta$ is the angle, in radians,

Question

[The area of a sector is $\frac{1}{2} r^2 \theta$, where $\theta$ is the angle, in radians, subtended by the arc at the centre of the circle of radius $r$.] In the diagram, $O$ is the centre of the circle with diameter $BC$ and radius $10$ cm. $A$ is the shaded area enclosed by the chord $CD$, the diameter $BC$ and the minor arc $BD$. Angle $BCD = x$ radians. Show that $A = 100x + 50 \sin 2x$.[3] If $x$ is increasing at a rate of $0.2$ radian/second, find the rate of change of $A$ when $x = \frac{\pi}{6}$.[3] Find the value of $x$ when the rate of change of $A$ is $50$ times the rate of increase of $x$.[3]

Accepted Answer

(a) $A=100x+50\sin2x$ (shown) (b) $30\text{ cm}^2\text{/s}$ (c) $x=\dfrac\pi3$ radians.

TKSS 4AM 2022 Prelims P2 Q7

TKSS 4AM 2022 Prelims P2 Q7

Solution:

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