Differentiation - Problems Involving Maximisation and Minimisation

The diagram shows an iron plate that consists of a quadrant and two identical rectangles, where the radius of the quadrant is \(x\) cm and the dimensions of the rectangles are \(x\) cm by \(y\) cm. Assume that the thickness of the plate is negligible and has a fixed total area of \(16\) \(\text{cm}^2\).

1. Show that the perimeter, \(P\) cm, of the plate, is given by \(P = 2x + \frac{32}{x}\).
2. Given that \(x\) can vary, find the values of \(x\) and \(y\) for which \(P\) has a minimum value.