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.                              
                                                                                                                                                                                                                                                                                                                                                                          
 Farmersville, California, 36° N, and Penticton, British Columbia, 49° N

Accepted Answer

Welcome back. I am so glad you're here. We are asked to calculate the distance between the following places on the assumption that they are on the same North south line, estimate the radius of the earth to be 6400 kilometers and express the final answer to the nearest kilometer. We've got Berlin Germany at 52 degrees north and Rome Italy at 42 degrees north. Our answer choices are answer choice. A 356 kilometers. Answer choice. B 1117 kilometers. Answer choice C 10,500 kilometers and answer choice D 3342 kilometers. All right. So if we imagine the earth, I'm going to draw a rough earth. Oh man, our earth looks like it's been impacted by a meteor that is not very spherical. But anyway, and we've got the earth and then imagining I'm drawing Europe. It's kind of like up in the first quadrant. If this was on top of, let's kind of imagine that it's on top of a coordinate plane. We've got north, south is our Y axis and then east west is our X axis. And so I'm drawing Europe up here in the first quadrant. And boy, this is a really rough Europe, but, you know, it's just going to be kind of blobby. There's Europe. What's important here is that on a north south line? So that would run all along the edge of the earth from the North pole to the south pole that I drew. We can envision Berlin Germany up here at 52 degrees north and then Rome Italy further south, it's no longer on a boot of a country here at 42 degrees north. And we can see that the distance we're measuring between them is actually an arc length. So here, the formula that we're going to use is for arc length, we recall from previous lessons and from the textbook that that is S equals R the and so R is going to be our radius. And we were given that the radius of the earth we're estimating is 6400 kilometers and theta is the angle measure, but it needs to be in radiance, not degrees. So here, if we were envisioning this is an angle, we can see that from 52 degrees north to 42 degrees north. Well, that's 10 degrees, but we need to convert that into radiance. So if we take 10 degrees, we recall from previous lessons to get from degrees to radiance, we're going to multiply that by the unit in the numerator, we have pi radiance that's divided by 180 degrees, the degrees units cancel out. And so that's going to be 10 pi divided by 180 radians, but 10 pi divided by 180 is going to simplify it to be just pi divided by 18 radiance. So our theta here, the one that we are allowed to use for arc length because it's in radiance is pi divided by 18. So now it's just plugging that into our arc length equation. So let's do that S equals instead of R, we have 6400 kilometers multiplied by R. Theta in radiance is pi divided by 18. And so multiplying this straight across, we have 6400 pi divided by 18. But they want it to the nearest, well, this is kilometers. They want it to the nearest kilometer, but not in terms of pi. So if you plug that into your calculator, you're going to get approximately 1117 kilometers. And that's about how far it is from Berlin to Rome. We look at our answer choices and that matches with answer choice. B well done. We'll catch you on the next one.

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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. Farmersville, California, 36° N, and Penticton, British Columbia, 49° N