Civil engineers design bridges to span over expressways. The diagram below represents a bridge over

Question

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 $x=a\left( 	heta -\sin 	heta ight)$, $y=a\left( 1-\cos 	heta ight)$, for $0\le 	heta \le 2\pi $, where $a$ is a positive constant. The units of $x$ and $y$ are metres. At the point $Q$, $	heta =\beta $ and at the point $R$, $	heta =2\pi -\beta $ Find, in terms of $a$ and $\beta $, the distance $PS$. [2] Show that the area of the shaded region on the diagram, representing the area under the bridge, is $\frac{1}{2}{{a}^{2}}\left( 6\pi -6\beta +8\sin \beta -\sin 2\beta ight)$ [6] It is given that the area under the bridge, in square metres, is $7.8159{{a}^{2}}$. Find the value of $\beta $. [1] The width of the expressway, $PS$ is $50$ metres. Find the greater and least heights of the arch, $QR$, above the expressway. [4]

Accepted Answer

(a) PS = 2a(π - β + sinβ) (b) Area = (1/2)a²(6π - 6β + 8sinβ - sin2β) (c) β = 1.90 (d) Least height = 15.1 m, Greatest height = 22.9 m

N2021 P1 Q11

N2021 P1 Q11

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