(a) Two places X and Y on the equator are on longitudes 67°E and 123°E respectively. (i) What is the distance between them along the equator? (ii) How far from the North pole is X? [Take (pi = frac{22}{7}) and radius of earth = 6400km].
(b) In the diagram, PQR is a circle centre O. N is the mid-point of chord PQ. |PQ| = 8cm, |ON| = 3cm and < ONR = 20°. Calculate the size of < ORN to the nearest degree.
Explanation
(a) Longitude difference = 123° – 67° = 56°
(i) Distance between X and Y along the equator = (frac{56}{360} times 2 times frac{22}{7} times 6400 cos 0)
= (frac{56}{360} times 2 times frac{22}{7} times 6400)
= (6257.78km approxeq 6258km)
(ii) Latitude difference = 90°
Distance from the North pole = (frac{90}{360} times 2 times frac{22}{7} times 6400)
= (10057.14km approxeq 10057km)
(b)
(OP^{2} = PN^{2} + ON^{2})
(4^{2} + 3^{2} = 25)
(OP = sqrt{25} = 5cm)
(OP = OR = 5cm)
(Delta NOR = scalene)
(frac{OR}{sin 20} = frac{3}{sin < ORN})
(frac{5}{sin 20} = frac{3}{sin < ORN})
(sin < ORN = frac{3 times sin 20}{5})
(sin < ORN = 0.2052)
(< ORN = sin^{-1} (0.2052))
(< ORN = 11.84° approxeq 12°)