(i) To find angle ABP, use the sine rule in triangle ABP:

\(\sin^{-1}\left(\frac{3}{5}\right) = 0.6435\) radians.

(ii) The area of a sector is given by \(\frac{1}{2} r^2 \theta\).

For sector BAP, \(r = 5\) and \(\theta = 0.6435\):

\(\text{Area} = \frac{1}{2} \times 5^2 \times 0.6435 = 8.04\).

For sector DAQ, \(r = 3\) and \(\theta = \frac{\pi}{2}\):

\(\text{Area} = \frac{1}{2} \times 3^2 \times \frac{\pi}{2} = 7.07\).

(iii) The area of the shaded region can be calculated by subtracting the area of triangle BPC from the sum of the areas of the sectors:

Area of triangle BPC = \(\frac{1}{2} \times 3 \times 4 = 6\).

Shaded area = \(8.04 + 7.07 - (15 - 6) = 6.11\).