David flies a drone with a flight path \(C\), modelled by the equation \(y=2^x\), where \(x \ge 0\) and \(y>0\). He stands at the origin, \(O\), and sees the drone at point \(P(a,b)\). Assuming David’s line of sight is the tangent to the curve \(C\) at point \(P\) which passes through the origin, find the exact coordinates of \(P\).[4]
In another instance, David stands at the origin \(O\) and flies the drone vertically upwards from a fixed point \(Q\), 500 m away from \(O\). The drone, represented by \(R\), is ascending upwards at a constant speed of 20 m/s. At the same time, David walks along the horizontal line segment \(OQ\) at a constant speed of 5 m/s. At any time \(t\) seconds, David’s position is point \(S\) on the horizontal line segment \(OQ\).
Given that the angle \(\theta\), in radians, is the angle \(SRQ\), show that \(\tan \theta=\dfrac{100-t}{4t}\).[1]
By implicit differentiation or otherwise, find \(\dfrac{d\theta}{dt}\) in terms of \(t\).[4]
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Answer:(a) \(P=\left(\dfrac1{\ln2},e\right)\) (b)(ii) \(\dfrac{\mathrm d\theta}{\mathrm dt}=-\dfrac{400}{16t^2+(100-t)^2}\text{ rad s}^{-1}\)