To calculate the distance between waypoints A and B using Mercator sailing, we can use the haversine formula. The formula calculates the distance between two points on a sphere given their latitudes and longitudes.
First, we need to convert the latitudes and longitudes from degrees, minutes, and seconds to decimal degrees:
Initial Position (A):
Latitude: 32°17.6’ S
Longitude: 007°14.6’ W
Convert the latitude to decimal degrees:
32 + 17.6/60 = 32.2933° S
Convert the longitude to decimal degrees:
-(7 + 14.6/60) = -7.2433° W (note the negative sign indicating west)
Final Position (B):
Latitude: 29°12.3’ S
Longitude: 001°38.4’ E
Convert the latitude to decimal degrees:
29 + 12.3/60 = 29.2050° S
Convert the longitude to decimal degrees:
1 + 38.4/60 = 1.6400° E
Now, we can use the haversine formula to calculate the distance between the two positions. However, since the initial position is located west of the final position, we need to adjust the longitude difference. For waypoints separated by more than 180 degrees of longitude, it is necessary to wrap the difference to the smaller angle between them:
Longitude difference adjusted = (360 - initial longitude) + final longitude = (360 - 7.2433) + 1.6400 = 354.3967°
Applying the haversine formula:
a = sin²((final latitude - initial latitude) / 2) + cos(initial latitude) * cos(final latitude) * sin²((longitude difference adjusted) / 2)
c = 2 * atan2(sqrt(a), sqrt(1 - a))
distance = radius of the Earth * c
Using the average radius of the Earth, which is approximately 6,371 km:
radius of the Earth = 6,371 km
a = sin²((29.2050 - 32.2933) / 2) + cos(32.2933) * cos(29.2050) * sin²(354.3967 / 2)
c = 2 * atan2(sqrt(a), sqrt(1 - a))
distance = 6,371 km * c
Calculating the values:
a = sin²((-3.0883) / 2) + cos(32.2933) * cos(29.2050) * sin²(354.3967 / 2)
≈ 0.0136
c = 2 * atan2(sqrt(0.0136), sqrt(1 - 0.0136))
≈ 0.1805
distance = 6,371 km * 0.1805
≈ 1,150 km
Therefore, the distance between waypoints A and B using Mercator sailing is approximately 1,150 kilometers.
Calculate the distance between these waypoints using Mercator sailing:
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
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