The diagram shows a cross-sectional area of a tunnel \(ABCDEF\). \(AB=4\) m, \(BC=10\) m and \(\angle BAF=\angle BCF=\theta\) where \(\theta\) is an angle in degrees. The tunnel is symmetrical about the vertical line \(CF\).
Show that \(AF=10\sin\theta+4\cos\theta\).[2]
Express \(AF\) in the form \(R\sin(\theta+\alpha)\) where \(R>0\) and \(0^\circ<\alpha<90^\circ\).[3]
State the maximum length of \(AF\) and the corresponding value of \(\theta\).[2]
The tunnel was built such that the length of \(AF\) is the maximum. Is it possible for a train of width 12 m and height 3.6 m to pass through the tunnel? You can assume that the train is a rectangular cuboid and that the train will pass through the center of the tunnel.[2]