2024 CGSS P1 Q10

The diagram shows a running track \(ABCD\). \(D\) is a point on the straight line \(EC\). It is given that \(AB = 280\text{ m}\), \(AD = 150\text{ m}\), \(\text{angle}\hspace{0.5em} BCD = \text{angle}\hspace{0.5em} DAB = 90^\circ\) and \(\text{angle}\hspace{0.5em} ADE = \theta\), where \(0^\circ < \theta < 90^\circ\).

1. Show that \(L\text{ m}\), the length of the running track, is given by
   \(L = p + 130\cos\theta + 430\sin\theta\), where \(p\) is a constant.[3]
2. Express \(L\) in the form \(p + R\cos(\theta - \alpha)\), where \(R > 0\) and \(\alpha\) is an acute angle.[3]
3. Xin Ting ran five complete laps around the track.
   She claims that she ran a total of \(5\text{ km}\). Explain why she might be wrong.[2]
4. Given that the running track is \(700\text{ m}\) long, find the value of \(\theta\).[3]