Surveying a coastal reserve

The diagram shows a coastal reserve bounded by four survey markers \(P\), \(Q\), \(R\) and \(S\) on level ground.

The bearing of \(P\) from \(S\) is \(342^\circ\) and \(SP=176\text{ m}\). Marker \(Q\) is \(80\text{ m}\) due west of \(P\).

The angle \(PQR\) is \(117^\circ\) and the bearing of \(P\) from \(R\) is \(052^\circ\).

Give non-exact numerical answers correct to \(3\) significant figures unless a different degree of accuracy is stated.

1. Find the bearing of \(R\) from \(P\).[1]
2. Find \(\angle QPR\).[1]
3. Show that the distance \(PR\) is \(168.66\text{ m}\), correct to two decimal places.[3]
4. Find the area of triangle \(PQR\).[2]
5. Find \(\angle RPS\).[1]
6. Hence, calculate the distance \(SR\).[3]
7. An observation tower, \(24\text{ m}\) high, stands vertically at \(P\).
   Find the angle of elevation of the top of the tower from \(S\).[2]