Answer :
To find the distance between two cities A and B on the Earth's surface, given an angle of separation [tex]\(\theta = 29^\circ\)[/tex] and the Earth's radius of approximately 4000 miles, we can use the formula for arc length. This formula is:
[tex]\[ \text{Distance} = \text{Radius} \times \theta \, (\text{in radians}) \][/tex]
Here is how you can calculate the distance step-by-step:
1. Convert the angle from degrees to radians:
- The formula to convert degrees to radians is:
[tex]\[
\text{radians} = \text{degrees} \times \frac{\pi}{180}
\][/tex]
- For [tex]\(\theta = 29^\circ\)[/tex]:
[tex]\[
\theta \, (\text{in radians}) = 29 \times \frac{\pi}{180} \approx 0.5061
\][/tex]
2. Calculate the arc length (distance between the two cities):
- Use the formula:
[tex]\[
\text{Distance} = 4000 \times 0.5061
\][/tex]
- This gives:
[tex]\[
\text{Distance} \approx 2024.58 \text{ miles}
\][/tex]
3. Round the distance to the nearest whole number:
- The rounded distance [tex]\(\approx 2025\)[/tex] miles.
Therefore, the distance between cities A and B is approximately 2025 miles.
[tex]\[ \text{Distance} = \text{Radius} \times \theta \, (\text{in radians}) \][/tex]
Here is how you can calculate the distance step-by-step:
1. Convert the angle from degrees to radians:
- The formula to convert degrees to radians is:
[tex]\[
\text{radians} = \text{degrees} \times \frac{\pi}{180}
\][/tex]
- For [tex]\(\theta = 29^\circ\)[/tex]:
[tex]\[
\theta \, (\text{in radians}) = 29 \times \frac{\pi}{180} \approx 0.5061
\][/tex]
2. Calculate the arc length (distance between the two cities):
- Use the formula:
[tex]\[
\text{Distance} = 4000 \times 0.5061
\][/tex]
- This gives:
[tex]\[
\text{Distance} \approx 2024.58 \text{ miles}
\][/tex]
3. Round the distance to the nearest whole number:
- The rounded distance [tex]\(\approx 2025\)[/tex] miles.
Therefore, the distance between cities A and B is approximately 2025 miles.
