2024 ACS(BR) P2 Q8

\(ABCD\) is a rectangle which fits inside a semicircle of radius \(8\text{ cm}\) and centre \(O\). It is given that \(AB=x\text{ cm}\) and \(BC=y\text{ cm}\).

1. Show that \(A\text{ cm}^2\), the area of the rectangle, is given by \(A=\dfrac x2\sqrt{256-x^2}\).[2]
2. Given that \(x\) can vary, find the value of \(x\) which gives a stationary value of \(A\).[4]
3. By considering the sign of \(\dfrac{\mathrm{d}A}{\mathrm{d}x}\), determine whether the stationary value of \(A\) is maximum or minimum.[2]