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.