To find the course (bearing) and distance between two points on the Earth's surface using the mean latitude method, we follow these steps:

Step 1: Convert the Coordinates to Decimal Degrees

First, convert the given coordinates from degrees and minutes to decimal degrees.

• Point A: 27° 15'N, 71° 23'W

  • Latitude A: $27° + \frac{15'}{60} = 27.25°N$
  • Longitude A: $71° + \frac{23'}{60} = 71.3833°W$

• Point B: 28° 22'N, 68° 18'W

  • Latitude B: $28° + \frac{22'}{60} = 28.3667°N$
  • Longitude B: $68° + \frac{18'}{60} = 68.3°W$

Step 2: Calculate the Mean Latitude

The mean latitude is the average of the latitudes of points A and B.

$\text{Mean Latitude} = \frac{\text{Latitude A} + \text{Latitude B}}{2} = \frac{27.25° + 28.3667°}{2} = 27.80835°N$

Step 3: Convert Mean Latitude to Radians

Since calculations often require radians, we convert the mean latitude to radians:

$\text{Mean Latitude in Radians} = 27.80835° \times \frac{\pi}{180} \approx 0.48512 \text{ radians}$

Step 4: Calculate the Differences in Latitude and Longitude

• Difference in Latitude ($\Delta \phi$):

$\Delta \phi = \text{Latitude B} - \text{Latitude A} = 28.3667° - 27.25° = 1.1167°$

• Difference in Longitude ($\Delta \lambda$):

$\Delta \lambda = \text{Longitude B} - \text{Longitude A} = 68.3° - 71.3833° = -3.0833°$

Step 5: Convert the Differences to Radians

• Difference in Latitude in radians: $\Delta \phi = 1.1167° \times \frac{\pi}{180} \approx 0.01949 \text{ radians}$

• Difference in Longitude in radians: $\Delta \lambda = -3.0833° \times \frac{\pi}{180} \approx -0.05382 \text{ radians}$

Step 6: Calculate the Distance

Using the formula for distance $D$:

$D = \sqrt{\left(\Delta \phi \cdot R\right)^2 + \left(\Delta \lambda \cdot R \cdot \cos(\text{Mean Latitude})\right)^2}$ where $R$ is the Earth's radius (approx. 6371 km).

Now substituting the values:

$D = \sqrt{\left(0.01949 \times 6371\right)^2 + \left(-0.05382 \times 6371 \times \cos(0.48512)\right)^2}$

We can calculate this step-by-step.

Step 7: Calculate the Course (Bearing)

The course or bearing $C$ can be found using the formula:

$C = \arctan\left(\frac{\Delta \lambda \cdot \cos(\text{Mean Latitude})}{\Delta \phi}\right)$

Final Calculation:

Let's compute the distance and bearing numerically.The distance between points A (27°15'N, 71°23'W) and B (28°22'N, 68°18'W) is approximately 327.69 kilometers. The course or bearing from point A to point B is approximately 67.73° (measured clockwise from the North).

Would you like further details or have any questions about the process? Here are some related questions to explore:

1. How does the mean latitude method compare to the great-circle distance method?
2. What is the significance of the Earth's curvature in calculating long distances?
3. Can you use this method for finding distances over longer ranges, like between continents?
4. What are the limitations of using the mean latitude method?
5. How would the result change if calculated using nautical miles instead of kilometers?

Tip: When working with bearings, always ensure that the final angle is normalized between 0° and 360° to avoid negative bearings.