2026 PCSS P2 Q6

A landscape architect is designing a new park as shown in the diagram above. The straight road \(AC\) runs due north-south. Point \(B\) is an existing entrance to the park. Point \(P\) is a proposed rest area on path \(AB\). Point \(Q\) is a water point on road \(AC\).

Given that \(AQ=8\text{ m}\), \(QC=20\text{ m}\), \(AB=30\text{ m}\), \(PQ=15\text{ m}\) and angle \(PAQ=65^\circ\).

1. Calculate bearing of \(B\) from \(A\).[1]
2. Find the length of the footpath \(BC\).[2]
3. The shortest footpath from \(P\) to road \(AC\) is to be constructed. Find the length of the shortest footpath.[4]
4. A lamp post \(BD\), 6 m tall, is to be installed vertically at point \(B\). Calculate the angle of elevation of the top of the lamp post, \(D\), from point \(C\).[2]