$O$ and $A$ are fixed points $200$m apart on a horizontal ground. A hobbyist, initially positioned a

Question

$O$ and $A$ are fixed points $200$m apart on a horizontal ground. A hobbyist, initially positioned at $O$, remotely released a drone (denoted by $B$) from $A$. As the drone ascends vertically above $A$ at a steady rate of $2$ m/s, the hobbyist (denoted by $P$) moves towards $A$ with a steady speed of $4$ m/s concurrently. At time $t$, the angle $APB$ is $	heta $ radians. Draw a clearly labelled diagram to show all the relevant information at time $t$, including distance $AB$ and $AP$ in terms of $t$. [1] By considering triangle $APB$, show that the rate of change of $	heta $ is given by $\frac{100}{{{t}^{2}}+{{\left( 100-2t ight)}^{2}}}$. [3] Using an algebraic method, find the time when the rate of change of $\frac{\mathrm{d}	heta }{\mathrm{d}t}$ is a maximum. [2] (You do not need to prove that maximum occurs at this time.)

Accepted Answer

(i) $OP = 4t$, $AB = 2t$; (ii) $\frac{d\theta}{\mathrm{d}t} = \frac{100}{t^2 + (100 - 2t)^2}$; (iii) $t = 40$ s

2018 RVHS Promo Q5

2018 RVHS Promo Q5

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