a research station in antartica is at approximately 75 degreess latitude. how many miles is the station from the equator, given that the radius of the earth is approximately 4000 miles? round your answer to the nearest mile.
Use Arc length = rθ ....(1)
where θ is the arc angle in radians.
Since θ=75°=75*180/π radians
the arc-length, or terrestrial distance can be calculated from equation (1).
To find the distance from the research station in Antarctica to the equator, we need to calculate the length of the arc on the Earth's surface between the two latitudes. Here's how you can do it:
1. Convert the latitude to radians: Divide the latitude by 180 degrees and multiply by π (pi). In this case, 75 degrees can be converted to radians using the formula: π * (75 / 180).
2. Calculate the length of the arc: Multiply the result from step 1 by the radius of the Earth. In this case, the length of the arc is approximately (π * (75 / 180)) * 4000 miles.
3. Round the answer to the nearest mile: Round the result obtained in step 2 to the nearest mile to get the final answer.
Let's calculate it:
Latitude in radians: π * (75 / 180) ≈ 1.30899694 radians
Length of the arc: (1.30899694) * 4000 miles ≈ 5235.98775 miles
Rounding to the nearest mile: 5235.98775 miles ≈ 5236 miles
So, the research station in Antarctica is approximately 5236 miles from the equator.