To solve this problem using Great Circle Sailing, we need to find the shortest distance between the two given positions A and B.
Step 1: Convert the given coordinates from degrees, minutes, and seconds to decimal degrees.
Initial Position, A:
Latitude: 11° 14' N = 11.233° N (approximately) [converted: 11 + 14/60]
Longitude: 125° 03' E = 125.05° E (approximately) [converted: 125 + 3/60]
Final Position, B:
Latitude: 08° 01' S = -8.017° S (approximately) [converted: -8 - 1/60]
Longitude: 079° 34' W = -79.567° W (approximately) [converted: -79 - 34/60]
Step 2: Use the haversine formula to calculate the great circle distance.
The haversine formula is given by:
d = 2r * arcsin(sqrt(sin²((Lat2 - Lat1) / 2) + cos(Lat1) * cos(Lat2) * sin²((Lon2 - Lon1) / 2)))
Where:
- d is the great circle distance
- r is the radius of the Earth (approximately 3440.07 nautical miles)
- Lat1 and Lat2 are the initial and final latitudes in radians
- Lon1 and Lon2 are the initial and final longitudes in radians
Let's calculate the great circle distance:
r = 3440.07 nautical miles
Lat1 = 11.233° N = 11.233 * (pi / 180) ≈ 0.1960 radians
Lon1 = 125.05° E = 125.05 * (pi / 180) ≈ 2.1822 radians
Lat2 = -8.017° S = -8.017 * (pi / 180) ≈ -0.1398 radians
Lon2 = -79.567° W = -79.567 * (pi / 180) ≈ -1.3872 radians
Using the haversine formula:
d = 2 * 3440.07 * arcsin(sqrt(sin²((0.1960 - (-0.1398))/2) + cos(0.1960) * cos(-0.1398) * sin²((-1.3872 - 2.1822)/2)))
Calculating the value inside the square root:
sin²((0.1960 - (-0.1398))/2) + cos(0.1960) * cos(-0.1398) * sin²((-1.3872 - 2.1822)/2) ≈ 0.000576
Using the value in the haversine formula:
d = 2 * 3440.07 * arcsin(sqrt(0.000576))
≈ 2 * 3440.07 * arcsin(0.024)
Using the arcsin value ≈ 0.024:
d ≈ 2 * 3440.07 * 0.024
≈ 165.61 nautical miles
Therefore, the Great Circle Distance between position A and B is approximately 165.61 nautical miles.
Solve the problem below using Great Circle Sailing
Calculate the Great Circle Distance (nautical miles)
Initial Position, A: (11° 14’ N, 125° 03’ E)
Final Position, B: (08° 01’ S, 079° 34’ W)
1 answer