2018 CJC Promo Q10
(b)
During the performance, a drone is positioned at the center of the circular stage, $O$, and a performer is standing at the circumference of the stage, $A$, as seen in the diagram above. The drone starts to ascend vertically at a constant speed of $3$ ms$^{-1}$ as the performer starts to walk towards $O$ with a constant speed of $2$ ms$^{-1}$. At time $t$, the drone is at point $P$ and the performer is at point $B$. It is given that the angle $OBP$ is $\alpha $ radians and the radius of the circular stage is $20$ m.
(i) Show that $\alpha ={{\tan }^{-1}}\left( \frac{3t}{20-2t} \right)$.
[1]
(ii) Find $\frac{\text{d}\alpha }{\text{d}t}$ in terms of $t$.
[3]
(iii) The drone will hover once the rate of change of $\alpha $ is at its maximum. State the value of $t$ at this instant.
[1]
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