(i) Use the Pythagorean theorem in triangle \(PQR\):

\(RS^2 = PR^2 - PQ^2 = 10^2 - 6^2\)

\(RS^2 = 100 - 36 = 64\)

\(RS = \sqrt{64} = 8 \text{ cm}\)

(ii) Use trigonometry to find \(\angle RPQ\):

\(\sin \theta = \frac{8}{10}\)

\(\theta = \arcsin\left(\frac{8}{10}\right) \approx 0.9273 \text{ radians}\)

(iii) The area of the shaded region is the area of the trapezium minus the areas of the two sectors:

Area of trapezium = \(40 \text{ cm}^2\)

Large sector area = \(\frac{1}{2} \times 8^2 \times 0.9273\)

Small sector angle = \(\pi - 0.9273\)

Small sector area = \(\frac{1}{2} \times 2^2 \times 2.214\)

Shaded area = \(40 - (\text{Large sector} + \text{Small sector}) = 5.90 \text{ cm}^2\)