2024 ACS(BR) P1 Q8

A civil engineer is designing a bridge which is \(131\) metres long, \(40\) metres high and is to have four identical parabolic arches along its length. Each arch is \(34\) metres high and there is one metre between bases of each adjacent pair of arches as shown in the diagram. A set of axes is placed with the origin at the left-hand end of the base of the first arch.

1. Find the \(x\)-intercepts of Arch 1.[2]
2. The equation representing Arch 1 can be written in the form \(y=a(x-p)(x-q)\). Show that \(a=-\dfrac{17}{128}\).[2]
3. Explain, with working, if the point \((50,30)\) lies on Arch 2.[3]