(i) To find the length of the arc DC, we first need to determine the radius of the circle with centre B. Using the Pythagorean theorem, we find \(r = \sqrt{72}\). The angle at the centre for the arc DC is \(\frac{\pi}{4}\) radians. The length of the arc is given by \(s = r \theta\), so \(s = \sqrt{72} \times \frac{\pi}{4} = \frac{3\sqrt{2}}{2} \pi \approx 6.66 \text{ cm}\).

(ii) The area of sector BDC is \(\frac{1}{2} \times 72 \times \frac{\pi}{4} = 9\pi\). The area of region Q is \(9\pi - 18\). The area of region P is \(\frac{1}{4} \pi \times 6^2 - \text{area of Q} = 18\). The ratio is \(\frac{18}{9\pi - 18} \approx 1.75\).