In this question all lengths are in centimetres.

The diagram shows a rectangle \(A B C D\) with \(B C=x\). The area of the rectangle is \(400 \mathrm{~cm}^{2}\). Two identical quarter-circles of radius \(\frac{x}{2}\), with centres \(A\) and \(C\), are removed from the rectangle to make the shaded shape.

Given that \(x\) can vary, find the value of \(x\) that gives the minimum value of the perimeter of the shaded shape and hence find this minimum value.