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\includegraphics[max width=\textwidth, alt={}, center]{b89c016e-dc56-48f4-b4c4-b432418e1b28-3_435_672_273_684} The diagram shows a curved rod \(A B\) of length 100 cm which forms an arc of a circle. The end points \(A\) and \(B\) of the rod are 99 cm apart. The circle has radius \(r \mathrm {~cm}\) and the arc \(A B\) subtends an angle of \(2 \alpha\) radians at \(O\), the centre of the circle.

1. Show that \(\alpha\) satisfies the equation \(\frac { 99 } { 100 } x = \sin x\).
2. Given that this equation has exactly one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\), verify by calculation that this root lies between 0.1 and 0.5.
3. Show that if the sequence of values given by the iterative formula $$x _ { n + 1 } = 50 \sin x _ { n } - 48.5 x _ { n }$$ converges, then it converges to a root of the equation in part (i).
4. Use this iterative formula, with initial value \(x _ { 1 } = 0.25\), to find \(\alpha\) correct to 3 decimal places, showing the result of each iteration.