2024 EJC H1 Q2

The diagram shows the layout of a garden. It comprises a rectangle \(PQRS\) and four identical semicircles with radius \(\frac{x}{2}\) m, two attached to \(QR\) and two attached to \(PS\). It is given that \(PQ = y\) m and the total perimeter of the garden is \(40\) m.

1. Show that the area, \(A\) m\({}^2\), of the garden is given by \(A = x \left( 40 - \frac{3\pi}{2}x \right)\).[3]
2. Use a non-calculator method to find the maximum value of \(A\), giving your answer in terms of \(\pi\). Justify that this is a maximum value.[4]