A civil engineer is designing a bridge which is 101 metres long, 30 metres high and is to have four…

A civil engineer is designing a bridge which is 101 metres long, 30 metres high and is to have four identical parabolic arches along its length. Each arch is 20 metres high and there is one metre between the base of each arch as shown in the diagram.

(a)
How wide is each arch at its base?
(b) A set of Cartesian axes is placed as shown in the diagram with the origin at the left hand end of the base of the first arch. Find the equation of the first arch. Use the general equation y = a(x  h)2 + k. (Hint − Substitute the x and y coordinates of known points on Arch 1 to determine the values of a, h and k.)
(c) Find the equation of the second arch.
(d) A river will flow through the third arch. Barges on the river are a standard size: 3 metres across and with the top of the barge 0.5 metres above the water. How far below the top of the arch can the water level rise and still allow a barge to pass through?

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