(i) We place the parabola so that the left side of the arch passes through (0, 0) and the right side will pass through (2, 0), since the arch is 2 m wide at the bottom. The vertex of the arch is at (1, 3).

General form of a parabola: y = ax²+ bx + c

At x = 0, y = 0 and on substituting, we get 0 = 0 + 0 + c. So c = 0.

At x = 1, y = 3 and on substituting, we get 3 = a + b.

At x = 2, y = 0 and on substituting, we get 0 = 4a + 2b.

This gives the simultaneous equations

a + b = 3

2a + b = 0

Subtracting the 1st line from the 2nd gives a = -3.

And so b = 6.

So the required parabola is y = -3x² + 6x, with x in metres.

The process of finding the equation is called modelling. It is a very important skill in science and engineering.

(ii) Now for the area: