2018 HCI Promo Q2

[It is given that the arc length of a circle is \(r\theta \), where \(r\) is the radius of the circle and \(\theta \) is the angle in radians subtended by the arc and the centre of the circle.]

The points \(A\left( 0,\,\,0.8 \right)\) and \(B\left( 0.8,\,\,0 \right)\) lie on the circumference of a circle with centre at the origin \(O\) and radius \(0.8\) units. Point \(P\) moves from \(A\), along the arc \(AB\) in the clockwise direction. Point \(Q\left( x,a \right)\) moves in the positive \(x\)-direction along the line \(y=a\), where \(a\) is a constant such that \(a>0.8\). At time \(t\) seconds, points \(P\) and \(Q\) move at a rate such that \(O\), \(P\) and \(Q\) are always on a line making an angle of \(\theta \) radians with the positive \(y\)-axis, where \(0\le \theta \le \frac{\pi }{2}\).

1. Show that point \(P\) is moving along the arc at a rate of \(0.8\frac{\mathrm{d}\theta }{\mathrm{d}t}\) units per second.[1]
2. Express \(\tan \theta \) in terms of \(x\) and \(a\). Given that \(\frac{\mathrm{d}x}{\mathrm{d}t}=0.6\) units per second, find in terms of \(a\) and \(x\), the rate at which the point \(P\) is moving along the arc.[4]