Question

To find the distance between two points on the earth when their latitude and longitude are known a right triangle can be used for an excellent approximation if the points are not too far apart. Point A is at latitude 42°30'45' N longitude 75°38'45" W. and point B is at latitude 42°31'30" N, longitude 75°39'30" W.

Find the distance from A to B in nautical miles. (One minute of latitude is one nautical mile.)

Answer 1

The distance between the two points on Earth, given in latitude and longitude, is approximately 0.934 nautical miles, determined using a right triangle approximation and adjusting for latitude.

Step-by-step explanation:

To find the distance between two points on the Earth using their latitude and longitude, we can use a simplified method if the two points are relatively close together, employing a right triangle approximation. We are given two points, A and B, with their respective coordinates, and asked to calculate the nautical mile distance between them. One crucial piece of information provided is that one minute of latitude corresponds to one nautical mile.

First, we determine the change in latitude and longitude between the two points. Point A is at latitude 42°30'45" N and longitude 75°38'45" W, while Point B is at latitude 42°31'30" N and longitude 75°39'30" W. To convert these coordinates from degrees, minutes, and seconds to minutes only (since one minute of latitude is equivalent to one nautical mile), we compute as follows:

1. Convert the change in latitude:
   42°31'30" - 42°30'45" = 0°00'45" or 0.75 minutes of latitude.
2. Convert the change in longitude:
   75°39'30" - 75°38'45" = 0°00'45" or 0.75 minutes of longitude.

However, we must consider that the distance covered by a minute of longitude varies with latitude, becoming smaller as one moves from the equator towards the poles. Since the latitude of the points is approximately 42°, we use a cosine factor to adjust the longitude difference. The approximate cosine of 42° is 0.743.

We then correct the longitude difference by multiplying 0.75 minutes by the cosine of the latitude:
0.75 * 0.743 ≈ 0.557 nautical miles of longitude difference.

Finally, to find the straight-line distance between Point A and Point B, we use the Pythagorean theorem for a right-angled triangle (since the problem suggests this method for an excellent approximation). The calculation is as follows:

Distance = √(latitude difference² + longitude difference²)

Distance = √(0.75² + 0.557²) ≈ √(0.563 + 0.310) ≈ √0.873 ≈ 0.934 nautical miles.

Therefore, the distance between Point A and Point B is approximately 0.934 nautical miles.