9758/2021/P1/Q11

Civil engineers design bridges to span over expressways. The diagram below represents a bridge over an expressway. $P S$.

In the diagram, $P Q$ and $S R$ are parallel to the $y$-axis, and $P Q=S R$. The arch of the bridge, $Q R$, forms part of the curve $O Q R C$ with parametric equations
$$x=a(\theta-\sin \theta), \quad y=a(1-\cos \theta), \quad \text { for } 0 \leqslant \theta \leqslant 2 \pi,$$
where $a$ is a positive constant. The units of $x$ and $y$ are metres.
At the point $Q, \theta=\beta$ and at the point $R, \theta=2 \pi-\beta$.

(a) Find, in terms of $a$ and $\beta$, the distance $P S$.



(b) Show that the area of the shaded region on the diagram, representing the area under the bridge, is
$$\frac{1}{2} a^2(6 \pi-6 \beta+8 \sin \beta-\sin 2 \beta) \text {. }$$



(c) It is given that the area under the bridge, in square metres, is $7.8159 a^2$. Find the value of $\beta$.



(d) The width of the expressway, $P S$, is 50 metres. Find the greatest and least heights of the arch, $Q R$, above the expressway.