Distance between Cities Find the distance in kilometers between e...

Question

Distance between Cities Find the distance in kilometers between each pair of cities, assuming they lie on the same north-south line. Assume the radius of Earth is 6400 km. See Example 2.                                                                                                                                                                                                                                                                                                                                                                                                         New York City, New York, 41° N, and Lima, Peru, 12° S

Accepted Answer

Hello, everyone. We are asked to calculate the distance between the following places. On the assumption that they're on the same north-south line. We want to estimate the radius of the earth to be 6400 kilometers. And we'll express the final answer to the nearest kilometer. We're given Buko Japan at 34 degrees north. And Darwin Australia at 12 degrees south. Our answer choices are a 782 kilometers. B 2457 kilometers C 5138 kilometers. D 1636 kilometers. First, we think of the earth as a circle. We can find the distance by using the arc length formula. So where s stands for arc length and that's gonna equal the radius multiplied by a central angle theta. So the distance we're looking for is going to be our arc length, our radius, we are told to estimate to 6400 kilometers. And our central angle we can find by taking the absolute value of the difference of our lines of latitude. So our latitude measures for the two cities, we're gonna take the absolute value of the distance. So Ukawa Japan is at 34 degrees north and we're going to subtract that from Darwin Australia, which since it is 12 degrees south, it, we're going to treat that as negative 12 because it would go below what we would think of as our X axis. So we're taking the absolute value of this, our double negatives are like adding. So we get the absolute value of 46 degrees, which is just 46 degrees. However, our formula, we need to have our, our central angle and radiance. So we need to take 46 degrees and convert it to radiance. We can do this by multiplying 46 by pi divided by 180. So when I do that and reduce, I get our central angle has a measure of 23 pi divided by 90 radians. So now we have all the information, we need to be able to use this formula to find our, our arc length. So S or arc length is gonna equal our radius 6400 multiplied by our central angle. So 23 pi divided by 90 and I'm going to reduce what I can and then multiply straight across the N reader and the denominator and it looks like it can reduce it by 10. And that will give me 14,725 divided by nine. So that's our arc length so far. However, we do not want to leave it in terms of pie. We want this rounded to the nearest kilometer. So we're gonna need to use a calculator. I'm going to multiply 14,720 by an estimation of pi and then divide by nine. And when I do so I get 5138.2493 kilometers. But our instructions wanted it in turn uh rounded to the nearest kilometer. So that means our arc length of s will be 5138 kilometers. And that's answer choice. C have a nice day.

Distance between Cities Find the distance in kilometers between each pair of cities, assuming they lie on the same north-south line. Assume the radius of Earth is 6400 km. See Example 2. New York City, New York, 41° N, and Lima, Peru, 12° S

Key Concepts

Great Circle Distance

Latitude

Earth's Radius

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