The diagram shows a sector \(OPQ\) of the circle centre \(O\), radius \(3r\) cm. The points \(S\) and \(R\) lie on \(OP\) and \(OQ\) respectively such that \(ORS\) is a sector of the circle centre \(O\), radius \(2r\) cm. The angle \(POQ=\theta\) radians. The perimeter of the shaded region \(PQRS\) is \(100\) cm.

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

(ii) Hence show that the area, \(A\text{ cm}^2\), of the shaded region \(PQRS\) is given by \(A=50r-r^2\).

(iii) Given that \(r\) can vary and that \(A\) has a maximum value, find this value of \(A\).

(iv) Given that \(A\) is increasing at the rate of \(3\text{ cm}^2\text{s}^{-1}\) when \(r=10\), find the corresponding rate of change of \(r\).

(v) Find the corresponding rate of change of \(\theta\) when \(r=10\).