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Answer :
To find the distance between two cities, [tex]\( A \)[/tex] and [tex]\( B \)[/tex], given that the angle [tex]\(\theta\)[/tex] between them is [tex]\(33^\circ\)[/tex] and the Earth's radius is approximately 4000 miles, we can follow these steps:
1. Understand the Concept: The distance between two points on the surface of a sphere (like Earth) can be found using the arc length formula. The arc length is the portion of the circumference of the circle that correlates with the given angle.
2. Convert Degrees to Radians: Since the arc length formula requires the angle in radians and we have the angle [tex]\(\theta = 33^\circ\)[/tex] in degrees, we first convert it to radians. The conversion formula is:
[tex]\[
\text{radians} = \text{degrees} \times \frac{\pi}{180}
\][/tex]
3. Apply the Formula: Once we have the angle in radians, the arc length [tex]\( s \)[/tex] (which is the distance between the two cities) can be calculated using the formula:
[tex]\[
s = r \times \theta_{\text{radians}}
\][/tex]
where [tex]\( r \)[/tex] is the radius of the Earth (4000 miles).
4. Calculate the Distance:
- Convert [tex]\(33^\circ\)[/tex] to radians:
[tex]\[
33^\circ \times \frac{\pi}{180} \approx 0.57596 \text{ radians}
\][/tex]
- Calculate the arc length (distance):
[tex]\[
s = 4000 \times 0.57596 \approx 2303.83 \text{ miles}
\][/tex]
5. Conclusion: Therefore, the distance between the two cities [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is approximately 2304 miles (rounded to the nearest mile).
1. Understand the Concept: The distance between two points on the surface of a sphere (like Earth) can be found using the arc length formula. The arc length is the portion of the circumference of the circle that correlates with the given angle.
2. Convert Degrees to Radians: Since the arc length formula requires the angle in radians and we have the angle [tex]\(\theta = 33^\circ\)[/tex] in degrees, we first convert it to radians. The conversion formula is:
[tex]\[
\text{radians} = \text{degrees} \times \frac{\pi}{180}
\][/tex]
3. Apply the Formula: Once we have the angle in radians, the arc length [tex]\( s \)[/tex] (which is the distance between the two cities) can be calculated using the formula:
[tex]\[
s = r \times \theta_{\text{radians}}
\][/tex]
where [tex]\( r \)[/tex] is the radius of the Earth (4000 miles).
4. Calculate the Distance:
- Convert [tex]\(33^\circ\)[/tex] to radians:
[tex]\[
33^\circ \times \frac{\pi}{180} \approx 0.57596 \text{ radians}
\][/tex]
- Calculate the arc length (distance):
[tex]\[
s = 4000 \times 0.57596 \approx 2303.83 \text{ miles}
\][/tex]
5. Conclusion: Therefore, the distance between the two cities [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is approximately 2304 miles (rounded to the nearest mile).
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