P and Q are two points on latitude 55°N and their longitudes are 33°W and 20°E respectively. Calculate the distance between P and Q measured along
(a) the parallel of latitude ;
(b) a great circle.
[Take (pi = frac{22}{7}) and radius of the earth = 6400km].
Explanation
(a) Angular difference = 33° + 23° = 56°
Distance PQ along parallel of latitude = (frac{56}{360} times 2 pi r)
where r = R cos 55°
(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° = R^{2} + R^{2} – 2R^{2} cos theta )
= (2r^{2} – 2r^{2} cos 56° = 2R^{2} – 2R^{2} cos theta)
But (r = R cos 55° = 6400 times 0.5736 = 3671)
(therefore 2r^{2} (1 – cos 56°) = 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°)
(therefore) Distance PQ along great circle = (frac{29.06}{360} times 2 times frac{22}{7} times 6400)
= (3247.34 km approxeq 3247km)