(a) Angular difference = 33^\circ + 23^\circ = 56^\circ Distance PQ along parallel of latitude = \(\frac{56}{360} \times 2 \pi r\) where r = R cos 55^\circ \(\therefore |PQ| = \frac{56}{360} \times 2 \times \frac{22}{7} \times 6400 \cos 55\) = \(3589.46 km \approxeq 3589 km\) (to the nearest degree) (b) \(|PO|^{2} = r^{2} + r^{2} - 2r^{2} \cos 56^\circ = R^{2} + R^{2} - 2R^{2} \cos \theta \) = \(2r^{2} - 2r^{2} \cos 56^\circ = 2R^{2} - 2R^{2} \cos \theta\) But \(r = R cos 55^\circ = 6400 \times 0.5736 = 3671\) \(\therefore 2r^{2} (1 - \cos 56^\circ) = 2R^{2} (1 - \cos \theta)\) \(2 \times 3671^{2} ( 1 - 0.5592) = 2 \times 6400^{2} (1 - \cos \theta)\) \(1 - \cos \theta = \frac{2 \times 3671^{2} \times 0.4408}{2 \times 6400^{2}}\) \(1 - \cos \theta = 0.1259\) \(\cos \theta = 1 - 0.1259 = 0.8741\) \(\theta = \cos^{-1} (0.8741) = 29.06^\circ\) \(\therefore\) Distance PQ along great circle = \(\frac{29.06}{360} \times 2 \times \frac{22}{7} \times 6400\) = \(3247.34 km \approxeq 3247km\)