Calculate the distance between these waypoints using Mercator sailing:

Question

Initial position (A) Lat 32°17.6’ S, Long 007°14.6’W
Final position (B)Lat 29°12.3’ S, Long 001°38.4’ E

Formulas:
Distance= DLAT/COS Course
Tan Co= DLO/DMP
MP= 7915.7logtan(lat2+45)-23.4sinlat

Answer 1

To calculate the distance between the waypoints using Mercator sailing, we will follow these steps:

1. Convert the latitude and longitude values from degrees, minutes, and seconds (DMS) to decimal degrees (DD).

For initial position (A):
Latitude: 32°17.6’ S = -32.2933°
Longitude: 007°14.6’ W = -7.2433°

For final position (B):
Latitude: 29°12.3’ S = -29.2050°
Longitude: 001°38.4’ E = 1.6400°

2. Calculate the difference in latitude (DLAT) and the difference in longitude (DLO) between the positions.

DLAT = Latitude(B) - Latitude(A) = (-29.2050°) - (-32.2933°) = 3.0883°
DLO = Longitude(B) - Longitude(A) = 1.6400° - (-7.2433°) = 8.8833°

3. Calculate the course (CO) using the formula:
CO = arctan(DLO / (DMP * cos(latitude(A))))

DMP: Distance to the Mercator Projection (10000 nautical miles)
Latitude(A) = -32.2933°

CO = arctan(8.8833 / (10000 * cos(-32.2933°)))

4. Calculate the distance (Distance) using the formula:
Distance = DLAT / cos(CO)

Distance = 3.0883° / cos(CO)

5. Substitute the CO value obtained in step 3 to calculate the final distance.

Alternatively, we can simplify the formulas using the values provided.

MP = 7915.7 * log(tan(latitude(B) + 45°)) - 23.4 * sin(latitude(B))

Latitude(B) = -29.2050°

MP = 7915.7 * log(tan(-29.2050° + 45°)) - 23.4 * sin(-29.2050°)