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

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.

Halifax, Nova Scotia , 45° N, and Buenos Aires, Argentina, 34° S

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. Our two places are Ottawa Canada at 45 degrees north and Santiago Chile at 34 degrees south. Our answer choices are answer choice. A 1229 kilometers. Answer choice B 391 kilometers. Answer choice C 2809 kilometers and answer choice D 8824 kilometers. All right. So I'm going to draw a really rough earth. You can imagine that it's a nice earth that we would all want to live on and not a squished egg like earth that I've drawn. And then I'm going to overlay that with our Y axis, that's going to be our north south and then our equator, that's our x axis, that's going to be our east west. And so we need to imagine Ottawa Canada. I'm going to put that in the second quadrant. I'm going to draw a blob of a North America. It doesn't look like North America. And then for Santiago Chile, I'm going to draw a blob of South America in the third quadrant, putting Ottawa Canada at 45 degrees north. And that's along that the left edge of what I drew for the circle for the earth. And then Santiago Chile at 34 degrees south in the third quadrant along that same edge. Because we're imagining that the north south line that they're on the same line of is the left half of what I drew for the earth. And we can see that the distance between them is an arc length. So we get to use the equation for an arc length. We recall from previous lessons, that would be S equals R theta R is our radius. We were given the radius of the earth 6400 kilometers. And now we just need to find theta theta is going to be that angle measure, but it needs to be in radiance. What we're given is in degrees. So first let's find it in degrees. We know that is going to be the distance from our 45 degrees north to 34 degrees south. So to find that distance will take the absolute value of the difference between the 245 degrees north is positive, that's above the X axis, finding the difference we're subtracting. But that 34 degrees south is negative 34 degrees because that's below the X axis. So essentially, we're adding 45 degrees plus 34 degrees, taking the absolute value of that and we get 79 degrees, but that's in degrees and we need to convert it to radians. So we recall from previous lessons, converting from degrees to radiance. We're going to multiply our degrees by in the numerator pi radiance that's divided by its equivalent 180 degrees and the degrees cancel out. And we have, we'll keep it as 79 pi divided by 180. And that's in radiance. So now we can plug in R and theta and figure out our arc length which is our distance. So for R that is 6400 kilometers multiplied by theta is 79 pi divided by 180. And for our final answer, we'll be expressing this to the nearest kilometer, not in terms of pi. So you can plug that into your calculator. The 6400 multiplied by 79 pi divided by 180. And you'll get when you round approximately 8824 and that's still in kilometers. And there's our approximate distance from Ottawa to Santiago. We look at our answer choices and this matches with answer choice. D well done. We'll catch you on the next one.

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.
Halifax, Nova Scotia , 45° N, and Buenos Aires, Argentina, 34° S

Key Concepts

Great Circle Distance on a Sphere

Latitude and Angular Distance

Arc Length Formula on a Sphere

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