Differentiation - Problems Involving Maximisation and Minimisation

A string of fixed length \(l\) m is cut into \(3\) segments to enclose a square, equilateral triangle and a circle, where the length of sides of the square and the equilateral triangle is \(x\) m and the radius of the circle is \(r\) m.

1. Express \(r\) in terms of \(x\) and \(l\).
2. Show that the total area of the \(3\) enclosed figures, \(A\) \(\text{m}^2\), is given by
   \(A = \left( \frac{\sqrt{3}}{4} + 1 \right)x^2 + \frac{(l - 7x)^2}{4\pi} .\)
3. Given that \(x\) can vary, find the value of \(x\) for which \(A\) has a minimum value.
4. Hence find the ratio of the lengths of the \(3\) segments cut out of the original string.