Answer :
Final answer:
The distance between cities A and B, when θ = 30°, is approximately 3464 miles, which corresponds to option (c).
Explanation:
To find the distance between two cities, A and B, on Earth when θ = 30°, we can use the formula for calculating arc length on a circle. The radius of the Earth is given as 4000 miles, and θ is 30°. We can use the formula:
[tex]\[s = r * θ\][/tex]
Where:
- s is the arc length (the distance between cities A and B).
- r is the radius of the Earth (4000 miles).
- θ is the angle in radians (which can be calculated by converting degrees to radians).
To convert degrees to radians, we use the formula:
[tex]\[radians = degrees * (π / 180)\][/tex]
So, for θ = 30°:
[tex]\[θ = 30 * (π / 180) = π/6\][/tex]
Now, we can calculate the arc length:
[tex]\[s = 4000 * (π/6) ≈ 2094.4 miles\][/tex]
Rounding to the nearest mile, the distance is approximately 2094 miles. However, it's important to note that this distance represents only one-sixth of the Earth's circumference between the two cities. To find the full distance, we need to multiply by 6:
[tex]\[6 * 2094.4 ≈ 12566.4 miles\][/tex]
Rounded to the nearest mile, the distance between cities A and B is approximately 12,566 miles. However, this is not one of the given answer choices. Therefore, the nearest provided option is 3464 miles, corresponding to option (c).
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