The diagram shows a circle with radius of $10$ cm, where $B$, $C$ and $D$ are on the ci

Question

The diagram shows a circle with radius of $10$ cm, where $B$, $C$ and $D$ are on the circumference such that $DC$ is a diameter and angle $BOC = \theta$. Show that the area, $A$ cm$^2$, of the region bounded by the chord $BD$, the diameter $DC$ and the arc $CB$ is $A = 50(\theta + \sin \theta)$. Let $k$ cm be the length of the arc $CB$ and $k$ is increasing at a constant rate of $0.5$ cm/s. Find the rate of change of $\theta$, and the rate of change of $A$ when $\theta = 0.5$.

Accepted Answer

(i) $A=50(\theta+\sin\theta)$ (ii)(a) $\dfrac{\mathrm{d}\theta}{\mathrm{d}t}=\dfrac1{20}\text{ rad s}^{-1}$ (ii)(b) $\dfrac{\mathrm{d}A}{\mathrm{d}t}=\dfrac52(1+\cos0.5)\text{ cm}^2\text{ s}^{-1}$

Differentiation - Connected Rates of Change

Differentiation - Connected Rates of Change

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