Question

calculating distance between two towns at different latitudes

Answer 1

To calculate the distance between two towns located at different latitudes and longitudes, you can use the haversine formula. This formula accounts for the curvature of the Earth and provides a reasonably accurate distance (great-circle distance) between two points specified in latitude and longitude.

### Haversine Formula
The formula is as follows:

\[
d = 2r \cdot \arcsin\left(\sqrt{\sin^2\left(\frac{\Delta \phi}{2}\right) + \cos(\phi_1) \cdot \cos(\phi_2) \cdot \sin^2\left(\frac{\Delta \lambda}{2}\right)}\right)
\]

Where:
- \( d \) is the distance between the two points (along the surface of the sphere).
- \( r \) is the radius of the Earth (mean radius = 6,371 km).
- \( \phi_1, \phi_2 \) are the latitudes of the two points in radians.
- \( \Delta \phi = \phi_2 - \phi_1 \) is the difference between the latitudes in radians.
- \( \Delta \lambda = \lambda_2 - \lambda_1 \) is the difference between the longitudes in radians.

### Steps to Calculate Distance
1. Convert the latitudes and longitudes from degrees to radians.
2. Apply the haversine formula to find the distance.

### Example Calculation
Let's say:
- Town A: Latitude 34.0522° N, Longitude 118.2437° W (Los Angeles)
- Town B: Latitude 40.7128° N, Longitude 74.0060° W (New York City)

1. Convert to radians:
- \( \phi_1 = 34.0522 \times \frac{\pi}{180} \)
- \( \phi_2 = 40.7128 \times \frac{\pi}{180} \)
- \( \lambda_1 = -118.2437 \times \frac{\pi}{180} \)
- \( \lambda_2 = -74.0060 \times \frac{\pi}{180} \)

2. Calculate differences:
- \( \Delta \phi = \phi_2 - \phi_1 \)
- \( \Delta \lambda = \lambda_2 - \lambda_1 \)

3. Apply the haversine formula to calculate \( d \).

### Sample Calculation:
1. Convert to radians (using a calculator):
- \( \phi_1 = 0.594323 \)
- \( \phi_2 = 0.710572 \)
- \( \lambda_1 = -2.063741 \)
- \( \lambda_2 = -1.291648 \)

2. Calculate differences:
- \( \Delta \phi = 0.710572 - 0.594323 = 0.116249 \)
- \( \Delta \lambda = -1.291648 - (-2.063741) = 0.772093 \)

3. Substitute values into the haversine formula:
\[
d = 2 \cdot 6371 \cdot \arcsin\left(\sqrt{\sin^2\left(0.116249/2\right) + \cos(0.594323) \cdot \cos(0.710572) \cdot \sin^2(0.772093/2)}\right)
\]

Calculating this step-by-step will give you the distance between the two towns.

For quicker calculation, you can also use online distance calculators or tools like Python libraries (e.g., `haversine` or `geopy`).