Given point B at latitude 12° N and longitude 110° W, and point A is 8008 km due south of B. Calculate the position of A. Then calculate the shorter distance in km between A and C along the parallel o

Here's how to solve the problem: a) Determine: a) (i) The position of A. Step 1: Calculate the difference in latitude () corresponding to the distance of 8008 km. The distance along a meridian is given by D = ()/(360^) × 2 R. Given D = 8008 km, R = 6370 km, and = (22)/(7). 8008 = ()/(360^) × 2 × (22)/(7) × 6370 8008 = ()/(360^) × (44)/(7) × 6370 8008 = ()/(360^) × 44 × 910 8008 = ()/(360^) × 40040 = (8008 × 360^)/(40040) = (8008 × 36^)/(4004) = 2 × 36^ = 72^ Step 2: Determine the latitude of A. Point B is at 12^ N. Point A is 8008 km south of B, so its latitude will be 12^ N - 72^. Latitude of A = 12^ - 72^ = -60^ This means the latitude of A is 60^ S. Since A is directly south of B, they share the same longitude. The longitude of B is 110^ W. Step 3: State the position of A. The position of A is (60^ S, 110^ W). a) (ii) The shorter distance in kilometres between A and C along the parallel of latitude. Step 1: Identify the coordinates of A and C. From (a)(i), A is (60^ S, 110^ W). Point C is on longitude 160^ E and on the same latitude as A. So, C is (60^ S, 160^ E). Step 2: Calculate the difference in longitude () between A and C. Since A is in the West and C is in the East, we add their longitudes: = 110^ + 160^ = 270^ For the shorter distance, we take the smaller angle around the Earth: Shorter = 360^ - 270^ = 90^ Step 3: Calculate the radius of the parallel of latitude (r). The latitude is = 60^. r = R = 6370 60^ r = 6370 × 0.5 = 3185 km Step 4: Calculate the distance between A and C along the parallel of latitude. The distance D_AC = Shorter 360^ × 2 r. D_AC = (90^)/(360^) × 2 × (22)/(7) × 3185 D_AC = (1)/(4) × 2 × (22)/(7) × 3185 D_AC = (1)/(2) × (22)/(7) × 3185 D_AC = (11)/(7) × 3185 D_AC = 11 × 455 D_AC = 5005 \