2024 IGCSE Additional Mathematics May/June 0606/21 Q11

In this question all lengths are in centimetres.

The diagram shows a rectangle \(ABCD\) with \(BC=x\). The area of the rectangle is \(400\,\mathrm{cm}^2\).

Two identical quarter-circles of radius \(\dfrac x2\), 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.[7]