A delivery truck beneath an arch

A bridge has a parabolic arch whose opening is \(8\text{ m}\) wide at road level and \(5\text{ m}\) high at its centre. The road is horizontal. Let \(y\) metres be the height of the arch above the road at a horizontal distance of \(x\) metres from the left end \(O\).

A delivery truck travels through the centre of the opening. Its front cross-section is modelled by a rectangle \(2.4\text{ m}\) wide and \(4.2\text{ m}\) high, including all parts of the truck. A minimum vertical clearance of \(0.25\text{ m}\) is required above every point on its roof.

1. The arch is modelled by \(y=a(x-4)^2+5\). Find \(a\).[2]
2. Calculate the smallest vertical clearance above the truck and determine whether it meets the requirement.[3]
3. The road is resurfaced, raising its level uniformly by \(0.12\text{ m}\). The arch is unchanged. Determine whether the same truck still meets the clearance requirement.[2]
4. Before resurfacing, find the greatest width of a rectangular truck of height \(4.2\text{ m}\) that can meet the clearance requirement when travelling through the centre. Give your answer correct to 3 significant figures.[3]