2 Fig. 6 shows the arch ABCD of a bridge.
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\caption{Fig. 6}
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The section from \(B\) to \(C\) is part of the curve \(O B C E\) with parametric equations
$$x = a ( \theta - \sin \theta ) , y = a ( 1 - \cos \theta ) \text { for } 0 \leqslant \theta \leqslant 2 \pi \text {, }$$
where \(a\) is a constant.
- Find, in terms of \(a\),
(A) the length of the straight line OE,
(B) the maximum height of the arch. - Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\).
The straight line sections AB and CD are inclined at \(30 ^ { \circ }\) to the horizontal, and are tangents to the curve at B and C respectively. BC is parallel to the \(x\)-axis. BF is parallel to the \(y\)-axis.
- Show that at the point B the parameter \(\theta\) satisfies the equation
$$\sin \theta = \frac { 1 } { \sqrt { 3 } } ( 1 \quad \cos \theta ) .$$
Verify that \(\theta = \frac { 2 } { 3 } \pi\) is a solution of this equation.
Hence show that \(\mathrm { BF } = \frac { 3 } { 2 } a\), and find OF in terms of \(a\), giving your answer exactly. - Find BC and AF in terms of \(a\).
Given that the straight line distance AD is 20 metres, calculate the value of \(a\).