2025 MI P1 Q6

A company produces customised badges. The diagram below shows the design of one such badge which consists of a rectangle and two identical semicircles.

It is given that \(AG = 2x\) cm, \(FG = y\) cm, \(BD = DF = x\) cm and the outer perimeter \(ABCDEFGA\) of the badge is \(25\) cm.

1. Show that the area of the badge, \(S\) cm\(^2\) is given by the formula \[S = 25x - 2x^2 - \frac{3}{4}\pi x^2.\][3]
2. Use calculus to find the exact value of \(x\) which gives the maximum value of \(S\).[4]