A and B are 2 points on 50°N. The two points lie on longitude 30°E and 150°W respectively. Calculate the distance in km from A to B along the parallel of latitude 6370.

To calculate the distance between two points on the same parallel of latitude, we can use the formula for the arc length of a circle. This is a common problem when dealing with geographic coordinates, and it uses the concept of great circles and parallels of latitude.

### Given Data:
- Latitude: \( 50^\circ N \)
- Longitude of Point A: \( 30^\circ E \)
- Longitude of Point B: \( 150^\circ W \)
- Radius of the Earth: \( 6370 \) km

### Step 1: Determine the Longitudinal Difference
Since Point A is at \( 30^\circ E \) and Point B is at \( 150^\circ W \), the difference in longitudes can be calculated as follows:

\[
\text{Difference in longitude} = 30^\circ + 150^\circ = 180^\circ
\]

### Step 2: Calculate the Effective Radius at Latitude 50°N
The Earth is not a perfect sphere, but for simplicity, we use the radius for a sphere of the Earth. The radius of the Earth at the equator is \( 6370 \) km, but this radius effectively decreases as we move away from the equator due to the cosine factor of the latitude.

The formula to calculate the effective radius at a given latitude is:

\[
R_{\text{effective}} = R_{\text{Earth}} \cdot \cos(\text{latitude})
\]

Here, the latitude is \( 50^\circ \):

\[
R_{\text{effective}} = 6370 \cdot \cos(50^\circ)
\]

Calculating \( \cos(50^\circ) \):

\[
\cos(50^\circ) \approx 0.6428
\]

Now substitute this value into the formula:

\[
R_{\text{effective}} = 6370 \cdot 0.6428 \approx 4094.636 \text{ km}
\]

### Step 3: Calculate the Arc Length (Distance)
The distance between the two points along the parallel is given by the arc length formula:

\[
\text{Arc length} = R_{\text{effective}} \cdot \left( \frac{\Delta \text{Longitude}}{360^\circ} \right) \cdot 2\pi
\]

Here, \(\Delta \text{Longitude} = 180^\circ\):

\[
\text{Distance} = 4094.636 \cdot \left( \frac{180}{360} \right) \cdot 2\pi
\]

\[
= 4094.636 \cdot \pi \text{ km}
\]

\[
\approx 4094.636 \cdot 3.1416
\]

\[
\approx 12863.7 \text{ km}
\]

### Final Answer
The distance from point A to point B along the 50°N parallel is approximately **12863.7 km**.

Would you like more details or have any questions?

### Related Questions:
1. How do you calculate the distance between two points along a great circle?
2. How does the Earth's curvature affect the calculation of distances between points?
3. What would be the distance if the latitude was 60°N instead of 50°N?
4. How does the cosine factor influence the effective radius at different latitudes?
5. What is the difference between traveling along a parallel of latitude versus a great circle?

### Tip:
Remember that the Earth's radius varies with latitude, being maximum at the equator and minimum at the poles. Use the cosine of the latitude to adjust for this variation.

Learn how to calculate the distance between two points A and B on the 50°N parallel using spherical geometry. This problem involves applying the arc length formula on a sphere, considering the Earth's curvature and latitude cosine adjustment. Discover step-by-step calculations and understand the impact of longitude differences on parallel distances.

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