2024 ASRJC H1 Q2

The diagram in Figure \(1\) shows a sign in the shape of a rectangle \(ABCD\) with a semicircle joined to \(CD\). The length of \(AB\) is \(2x\) cm. The total perimeter of the sign is \(50\) cm.

1. Show that the area of the sign in Figure \(1\) is \(50x - 2x^2 - \frac{\pi x^2}{2}\).[2]

An equilateral triangle with the side of \(0.25x\) cm is to be removed from the sign as shown in Figure \(2\). The area of the new sign is to be as large as possible.

2. Use differentiation to find the value of \(x\) that will give the maximum area of the new sign.[5]