During the performance, a drone is positioned at the center of the circular stage, $O$, and a perfor

Question

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. Show that $\alpha ={{	an }^{-1}}\left( \frac{3t}{20-2t} ight)$. [1] Find $\frac{\mathrm{d}\alpha }{\mathrm{d}t}$ in terms of $t$. [3] The drone will hover once the rate of change of $\alpha $ is at its maximum. State the value of $t$ at this instant. [1]

Accepted Answer

$\alpha = \tan^{-1}\left(\frac{3t}{20 - 2t}\right)$; $\frac{d\alpha}{dt} = \frac{60}{13t^2 - 80t + 400}$; maximum at $t = \frac{40}{13}$ s

2018 CJC Promo Q10

2018 CJC Promo Q10

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