To solve the problem of finding the distance between City A and City B, we need to use the concept of arc length on the surface of the Earth. Given that the radius of the Earth is 3960 miles, and that City A and City B are located at specific latitudes, we can follow these steps:
1. Convert Latitude from Degrees and Minutes to Decimal Degrees:
- Latitude of City A: 48°27' N
- Latitude of City B: 42°45' N
To convert from degrees and minutes to decimal degrees:
- City A: \(48 + \frac{27}{60}\)
- City B: \(42 + \frac{45}{60}\)
This gives us:
- City A: \(48.45\) degrees
- City B: \(42.75\) degrees
2. Convert the Latitudes from Degrees to Radians:
Latitude needs to be in radians when dealing with angular measurements for arc lengths. The formula to convert degrees to radians is:
[tex]\[
\text{radians} = \text{degrees} \times \frac{\pi}{180}
\][/tex]
Thus, calculating the latitudes in radians:
- City A: \(48.45 \times \frac{\pi}{180}\)
- City B: \(42.75 \times \frac{\pi}{180}\)
This results in:
- City A: \(0.845708\) radians (approximately)
- City B: \(0.746224\) radians (approximately)
3. Find the Difference in Latitude:
[tex]\[
\Delta \text{latitude} = \text{Latitude of City A (in radians)} - \text{Latitude of City B (in radians)}
\][/tex]
[tex]\[
\Delta \text{latitude} = 0.845708 - 0.746224 = 0.099484 \text{ radians} (approximately)
\][/tex]
4. Calculate the Distance Using Arc Length Formula:
The formula for the arc length \(s\) is:
[tex]\[
s = \theta \times r
\][/tex]
where \(\theta\) is the central angle in radians, and \(r\) is the radius of the Earth.
Thus,
[tex]\[
s = 0.099484 \times 3960
\][/tex]
This gives us:
[tex]\[
s \approx 393.956 \text{ miles}
\][/tex]
When we round 393.956 to the nearest mile, we get:
[tex]\[
s \approx 394 \text{ miles}
\][/tex]
Therefore, the distance between City A (48°27' N) and City B (42°45' N) is approximately 394 miles.
