a) Calculate the shortest distance between A and B along their parallel of latitude, correct to 3 significant figures.

Here's the solution to your problem. We'll use the Earth's radius R = 6371 km. a) Calculate the shortest distance between A and B along their parallel of latitude, correct to 3 significant figures. Town A is at (36^ N, 80^ E) and Town B is at (36^ N, 25^ E). Both towns are on the same latitude, = 36^ N. The difference in longitude is 80^ - 25^ = 55^. Step 1: Apply the formula for distance along a parallel of latitude. The formula is D = ()/(360^) × 2 R . D_AB = (55^)/(360^) × 2 (6371 km) (36^) Step 2: Calculate the numerical value. D_AB = (55)/(360) × 2 (6371) × 0.80901699 D_AB = 0.152777... × 2 (6371) × 0.80901699 D_AB ≈ 5219.09 km Step 3: Round to 3 significant figures. D_AB ≈ 5220 km The shortest distance between A and B is 5220 km. b) What percentage of the equator is the parallel of latitude 36^? Correct to the nearest whole number. Step 1: Use the relationship between the circumference of a parallel of latitude and the equator. The circumference of a parallel of latitude at angle is C_lat = 2 R . The circumference of the equator is C_eq = 2 R. The percentage is C_latC_eq × 100\% = (2 R )/(2 R) × 100\% = × 100\%. Here, = 36^. Step 2: Calculate the percentage. Percentage = (36^) × 100\% Percentage ≈ 0.80901699 × 100\% Percentage ≈ 80.901699\% Step 3: Round to the nearest whole number. Percentage ≈ 81\% The parallel of latitude 36^ is approximately 81% of the equator. c) P is a point due south of B. The distance along the great circle through B and P is 12,500 km. Calculate, correct to the nearest degree, the latitude of P. Town B is at (36^ N, 25^ E). Point P is due south of B, so its longitude is also 25^ E. The distance along a great circle (meridian) is D = ()/(360^) × 2 R, where is the difference in latitude. Given D = 12,500 km. Step 1: Calculate the difference in latitude . 12500 km = ()/(360^) × 2 (6371 km) = (12500 × 360^)/(2 × 6371) = (4500000)/(12742) ≈ 112.50^ Step 2: Determine the latitude of P. Since P is due south of B, its latitude will be 36^ N - . Latitude of P = 36^ - 112.50^ Latitude of P = -76.50^ A negative latitude indicates the Southern Hemisphere. Step 3: Round to the nearest degree. Latitude of P ≈ -77^ So, the latitude of P is 77^ S. The latitude of P is 77^ S. That's 2 down. 3 left today — send the next one.