N2021 P1 Q11

Civil engineers design bridges to span over expressways. The diagram below represents a bridge over an expressway, \(PS\).

In the diagram, \(PQ\) and \(SR\) are parallel to the \(y\)-axis, and \(PQ=SR\). The arch of the bridge, \(QR\), forms part of the curve \(OQRC\) with parametric equations

\(x=a\left( \theta -\sin \theta \right)\), \(y=a\left( 1-\cos \theta \right)\), for \(0\le \theta \le 2\pi \),

where \(a\) is a positive constant. The units of \(x\) and \(y\) are metres.

At the point \(Q\), \(\theta =\beta \) and at the point \(R\), \(\theta =2\pi -\beta \)

1. Find, in terms of \(a\) and \(\beta \), the distance \(PS\).[2]
2. Show that the area of the shaded region on the diagram, representing the area under the bridge, is
   \(\frac{1}{2}{{a}^{2}}\left( 6\pi -6\beta +8\sin \beta -\sin 2\beta \right)\)[6]
3. It is given that the area under the bridge, in square metres, is \(7.8159{{a}^{2}}\). Find the value of \(\beta \).[1]
4. The width of the expressway, \(PS\) is \(50\) metres. Find the greater and least heights of the arch, \(QR\), above the expressway.[4]