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Answer :
To find the distance between two cities, A and B, given that the central angle [tex]\(\theta\)[/tex] is [tex]\(25^{\circ}\)[/tex] and the radius of the Earth is approximately 4000 miles, you can follow these steps:
1. Understand the Formula: The formula to find the arc length (distance) between two cities, when you know the central angle in degrees and the radius, is:
[tex]\[
\text{Distance} = \theta \times \frac{\pi}{180} \times \text{Radius}
\][/tex]
This formula converts the angle from degrees to radians and then calculates the arc length.
2. Convert the Angle from Degrees to Radians: Since the formula requires the angle [tex]\(\theta\)[/tex] to be in radians, convert [tex]\(25^{\circ}\)[/tex] to radians:
[tex]\[
\theta_{\text{radians}} = 25 \times \frac{\pi}{180}
\][/tex]
Simplifying this calculation:
[tex]\[
\theta_{\text{radians}} \approx 0.436 \text{ radians}
\][/tex]
3. Calculate the Distance: Using the converted angle in radians, calculate the arc length:
[tex]\[
\text{Distance} = 0.436 \times 4000
\][/tex]
Doing this multiplication gives:
[tex]\[
\text{Distance} \approx 1745.33 \text{ miles}
\][/tex]
4. Round the Result: Round the distance to the nearest whole number:
[tex]\[
\text{Distance} \approx 1745 \text{ miles}
\][/tex]
Thus, the distance between the two cities, A and B, is approximately 1745 miles.
1. Understand the Formula: The formula to find the arc length (distance) between two cities, when you know the central angle in degrees and the radius, is:
[tex]\[
\text{Distance} = \theta \times \frac{\pi}{180} \times \text{Radius}
\][/tex]
This formula converts the angle from degrees to radians and then calculates the arc length.
2. Convert the Angle from Degrees to Radians: Since the formula requires the angle [tex]\(\theta\)[/tex] to be in radians, convert [tex]\(25^{\circ}\)[/tex] to radians:
[tex]\[
\theta_{\text{radians}} = 25 \times \frac{\pi}{180}
\][/tex]
Simplifying this calculation:
[tex]\[
\theta_{\text{radians}} \approx 0.436 \text{ radians}
\][/tex]
3. Calculate the Distance: Using the converted angle in radians, calculate the arc length:
[tex]\[
\text{Distance} = 0.436 \times 4000
\][/tex]
Doing this multiplication gives:
[tex]\[
\text{Distance} \approx 1745.33 \text{ miles}
\][/tex]
4. Round the Result: Round the distance to the nearest whole number:
[tex]\[
\text{Distance} \approx 1745 \text{ miles}
\][/tex]
Thus, the distance between the two cities, A and B, is approximately 1745 miles.
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