Robyn H.

asked • 02/05/15

The latitude of Athens is 37.97 degrees North and the latitude of Hammerfest is 70.63 degrees North. Find the distance in between them.

PreCalculus Test Question.

Mark M.

Are the two cities on the same longitude? What is the circumference of the earth?
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02/05/15

Robyn H.

This was the actual question, it wouldn't let me put it all initially:
 
Find the distance between Athens, Greece and Hammerfest, Norway, which lie on the same longitude.  The latitude of Athens is 37.97 degrees north and the latitude of Hammerfest is 70.63 degrees north.  Use a radius of 3960 miles for the Earth and round your solution to two decimal places.
 
We are just learning the beginning stages of trigonometry, haven't learned vectors, etc.
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02/08/15

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Russ P. answered • 02/07/15

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Robyn,
 
Athens, Greece you say is at Latitude is 37.97o N .  I got 37o58' N and Longitude 23o42' East. That's close.
Hammerfest, Norway you say at latitude 70.63o N. I got 62o00' N and Longitude 10o00' East. That's a big difference??
Earth's circumference is 40,075 km or 24,900 miles. 
 
Now there is a quick & dirty way of calculating it using plane geometry, or a more accurate method using vectors from the center of the Earth to each city, taking their dot product to get the included angle and using that in a ratio to get the arc length on the curved surface of the Earth.  I'll take the easy way using my data.
 
The Latitude difference is 62o00' - 37o58' = 24o02'N = 24.033 oN
Then the Latitude arc length X is X/24900 = 24.033/ 360.00  or  X = 1,662 miles
 
The Longitude difference is 23o42' - 10o00' = 13o42' E = 13.70 o E
Then the Longitude arc length Y is Y/24900 = 13.70/360        or Y = 1,155 miles
 
Forgetting the curvature of the spherical Earth, you form a right triangle with those leg distances. Then its hypotenuse length is 
 
c2 = a2 + b2  = (1662)2 + (1155)2 = 2,762,244 +  1,334,025  = 4,096,269
c  =  2,024 miles approximately as the distance between these two cities.
 
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Russ P.

Robyn : My response to your comment.
 
They really simplified your problem by stating that the two cities are assumed to be on the same longitude.
 
So assume that a spherical orange is like the Earth. Slice it in half through its poles and and look at the inside surface of the orange. Its perimeter is a perfect circle. It represents the longitude meridian circle that cuts through both cities.
 
Given this circle, you no longer need to deal with spheres and triangles. It now becomes a proportion problem on the circle’s perimeter that you can solve. I’ll give you some tips and the answer.
 
Let the horizontal line through the circle’s center be the x-axis from which you will measure latitude angles and the circle’s radius R. Spot the latitude angles on the circle’s rim (or perimeter) for both cities, and take the difference of the latitudes. The arc on the circle’s perimeter between those two cities is the distance you need to compute., and the angle between the latitudes is the angular measure of that distance.
 
If you start at the x-axis and sweep the angle through the full 360 degrees of a circle, then distance would be the circumference of the circle, C = 2(3.14159)R.
 
But the angle between the 2 cities is much smaller than 360 degrees. It’s actually only 0.0907 of 360 degrees as a decimal fraction.
 
So now you know everything to set up the correct proportion to compute the unknown arc length. Hint, it comes out to be about 2,257 miles.
 
So work the problem because you’ll see something like it on a test.
 
Russ
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02/08/15

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