A ship passing a lighthouse within radar range

A ship leaves port \(P\) and sails \(12\) km on a bearing of \(060^\circ\) to \(A\), then \(9\) km on a bearing of \(150^\circ\) to \(B\). A lighthouse \(L\) is \(10\) km due east of \(P\). Assume the sea is flat and each leg is a straight line.

1. Show that \(\angle PAB=90^\circ\). Hence find the distance \(PB\) and the bearing of \(B\) from \(P\).[4]
2. Find the distance \(AL\) and the angle \(PAL\).[3]
3. The lighthouse's radar detects ships within \(5.5\) km. Find the shortest distance between the ship and \(L\) while the ship sails from \(A\) to \(B\), and state whether the ship is detected on this leg.[3]