2026 NORTHBROOKS P1 Q11

The diagram shows a rectangular cargo ship \(ABCD\) measuring 50 m by 30 m moving down a river. The cargo ship is stuck such that point \(C\) is in contact with one side of the river and the points \(C\) and \(F\) form a straight line. Point \(A\) is in contact with the opposite side of the river where the points \(A\) and \(E\) form a straight line. It is given that angle \(DAE=\text{angle}\hspace{0.5em}BCF=\theta\), where \(0^\circ<\theta<90^\circ\).

1. Show that the width of the river is given by \(W=50\sin\theta+30\cos\theta\).[2]
2. Express \(W\) in the form \(R\sin(\theta+\alpha)\), where \(R>0\) and \(0^\circ<\alpha<90^\circ\).[3]
3. Find the maximum possible value of \(\theta\) if the cargo ship wishes to pass through a river of width 40 m.[2]