In this question, all lengths are in metres.

The diagram shows a window formed by a semi-circle of radius \(r\) on top of a rectangle with dimensions \(2r\) by \(y\). The total perimeter of the window is \(5\).

(i) Find \(y\) in terms of \(r\).

(ii) Show that the total area of the window is

\(A=5r-\frac{\pi r^2}{2}-2r^2.\)

(iii) Given that \(r\) can vary, find the value of \(r\) which gives a maximum area of the window and find this area. You are not required to show that this area is a maximum.