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Answer :
To find the distance between two cities, A and B, given that [tex]\(\theta = 28^\circ\)[/tex] and the radius of the Earth is approximately 4000 miles, we can use the concept of arc length on a circle.
Here’s a step-by-step guide to solving this:
1. Understand the Problem:
- We need to find the arc length on the surface of the Earth, which represents the distance between the two cities.
- The Earth can be thought of as a circle with a radius of 4000 miles.
2. Convert the Angle:
- The angle [tex]\(\theta = 28^\circ\)[/tex] is given in degrees.
- For arc length calculations, angles are usually in radians, so we need to convert degrees to radians.
- The conversion factor is: [tex]\(1^\circ = \frac{\pi}{180}\)[/tex] radians.
- Thus, [tex]\(28^\circ\)[/tex] in radians is [tex]\(28 \times \frac{\pi}{180}\)[/tex].
3. Calculate the Arc Length:
- The formula for arc length [tex]\(s\)[/tex] is:
[tex]\[
s = r \times \theta
\][/tex]
where [tex]\(r\)[/tex] is the radius and [tex]\(\theta\)[/tex] is in radians.
- Here, [tex]\(r = 4000\)[/tex] miles and [tex]\(\theta\)[/tex] is the radian value we calculated.
4. Perform the Calculation:
- Using the arc length formula, calculate [tex]\(s\)[/tex] by:
[tex]\[
s = 4000 \times 28 \times \frac{\pi}{180}
\][/tex]
- Compute this value to get the distance.
5. Round the Result:
- Finally, round the calculated distance to the nearest whole mile.
After performing these calculations, we find that the distance between the two cities, A and B, is approximately 1955 miles.
Here’s a step-by-step guide to solving this:
1. Understand the Problem:
- We need to find the arc length on the surface of the Earth, which represents the distance between the two cities.
- The Earth can be thought of as a circle with a radius of 4000 miles.
2. Convert the Angle:
- The angle [tex]\(\theta = 28^\circ\)[/tex] is given in degrees.
- For arc length calculations, angles are usually in radians, so we need to convert degrees to radians.
- The conversion factor is: [tex]\(1^\circ = \frac{\pi}{180}\)[/tex] radians.
- Thus, [tex]\(28^\circ\)[/tex] in radians is [tex]\(28 \times \frac{\pi}{180}\)[/tex].
3. Calculate the Arc Length:
- The formula for arc length [tex]\(s\)[/tex] is:
[tex]\[
s = r \times \theta
\][/tex]
where [tex]\(r\)[/tex] is the radius and [tex]\(\theta\)[/tex] is in radians.
- Here, [tex]\(r = 4000\)[/tex] miles and [tex]\(\theta\)[/tex] is the radian value we calculated.
4. Perform the Calculation:
- Using the arc length formula, calculate [tex]\(s\)[/tex] by:
[tex]\[
s = 4000 \times 28 \times \frac{\pi}{180}
\][/tex]
- Compute this value to get the distance.
5. Round the Result:
- Finally, round the calculated distance to the nearest whole mile.
After performing these calculations, we find that the distance between the two cities, A and B, is approximately 1955 miles.
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