Question

Calculate the distance between these waypoints using Mercator sailing:
Distance= Dlat/ Cos course
Tan Co= DLO/DMP
MP= 79.15.logtan7(lat/2+45)-23.4sinlat

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

Answer 1

To calculate the distance between waypoints A and B using Mercator sailing, we need to determine the following values:

A: Initial position
A_lat = 32°17.6' S
A_long = 007°14.6' W

B: Final position
B_lat = 29°12.3' S
B_long = 001°38.4' E

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

A_lat_DD = -32.2933° (approx.)
A_long_DD = -7.2433° (approx.)

B_lat_DD = -29.2050° (approx.)
B_long_DD = 1.6400° (approx.)

2. Calculate the difference in latitude (Dlat) and difference in longitude (DLO):

Dlat = B_lat_DD - A_lat_DD = -2.0883° (approx.)
DLO = B_long_DD - A_long_DD = 8.8833° (approx.)

3. Convert the course angle from degrees to radians:

Course = atan(DLO / Dlat) = atan(8.8833° / -2.0883°) ≈ -1.3330 radians

4. Calculate the distance (DMP) using the given formula:

MP = 79.15 * log(tan((A_lat_DD / 2) + 45°)) - 23.4 * sin(A_lat_DD)
MP ≈ 79.15 * log(tan((-32.2933° / 2) + 45°)) - 23.4 * sin(-32.2933°)
MP ≈ 79.15 * log(tan(56.8533°)) - 23.4 * sin(-32.2933°)
MP ≈ 79.15 * log(1.3586) - 23.4 * sin(-32.2933°)
MP ≈ 79.15 * 0.3098 - (-16.3569)
MP ≈ 24.5 + 16.3569
MP ≈ 40.8569

Distance = Dlat / cos(Course) = -2.0883° / cos(-1.3330 radians)
Distance = -2.0883° / cos(-1.3330)
Distance ≈ 1.7304

The distance between waypoints A and B using Mercator sailing is approximately 1.7304 nautical miles.