(i) The area of the semicircle is \(\frac{1}{2} \pi r^2\). The area of the sector BOC is \(\frac{1}{2} r^2 x\). The area of the triangle BOC is \(\frac{1}{2} r^2 \sin x\). The area of the segment is \(\frac{1}{2} r^2 (x - \sin x)\). Given that this is a quarter of the semicircle's area, we have:

\(\frac{1}{2} r^2 (x - \sin x) = \frac{1}{4} \times \frac{1}{2} \pi r^2\)

\(x - \sin x = \frac{1}{4} \pi\)

\(x = \frac{3}{4} \pi - \sin x\)

(ii) Evaluate \(f(x) = x - (\frac{3}{4} \pi - \sin x)\) at \(x = 1.3\) and \(x = 1.5\):

\(f(1.3) = 1.3 - (\frac{3}{4} \pi - \sin 1.3) \approx -0.047\)

\(f(1.5) = 1.5 - (\frac{3}{4} \pi - \sin 1.5) \approx 0.155\)

Since \(f(1.3) < 0\) and \(f(1.5) > 0\), there is a root between 1.3 and 1.5.

(iii) Using the iterative formula:

\(x_1 = \frac{3}{4} \pi - \sin 1.4 \approx 1.3805\)

\(x_2 = \frac{3}{4} \pi - \sin 1.3805 \approx 1.3793\)

\(x_3 = \frac{3}{4} \pi - \sin 1.3793 \approx 1.3796\)

\(x_4 = \frac{3}{4} \pi - \sin 1.3796 \approx 1.3795\)

The root correct to 2 decimal places is 1.38.