Tipping a crate under a low beam

A crate with a rectangular cross-section, \(4\) m wide and \(3\) m tall, is tipped about one bottom edge through an angle \(\theta\), where \(0^\circ\le\theta\le90^\circ\). The height of its highest corner above the floor is then \(h=3\cos\theta+4\sin\theta\) metres. Assume the floor is level and the crate is rigid.

1. Express \(h\) in the form \(R\cos(\theta-\alpha)\), where \(R>0\) and \(0^\circ<\alpha<90^\circ\).[3]
2. Find the greatest height of the highest corner and the angle at which it occurs.[2]
3. The crate is tipped from \(\theta=0^\circ\) to \(\theta=90^\circ\) under a beam \(4.5\) m above the floor. Find the range of values of \(\theta\) for which the crate would hit the beam, and comment on what this means for the move.[3]