Astronomy Rotation Problem?

Question

How far km is 35 degrees north from 35 degrees south at the same longitude?

HELP!

Answer 1

7784 km

Answer 2

First approximation, considering Earth to be spherical:

Using degrees.
A nautical mile is a minute (1/60 of a degree) of latitude. The distance from 35 N to 35 S is 70 degrees = 4200 minutes = 4200 nautical miles. There is 0.54 nautical mile in one km.
4200 n.m. / 0.54 nm per km = 7777.77... km.

Using grades.
There are 90 degrees in a right angle.
There are 100 grades in a right angle.
Therefore, 1 degree is 10/9 grade.
70 degrees = 70*10/9 = 77.7777... grades.
There are 100 centigrade in one grade, and one centigrade of latitude on the surface of the Earth is one km.
77.77777... grades = 7777.77... centigrades = 7777.77... km.

If you consider the Earth to be an ellipsoid with a "compression" of 1/293.465 (closer to the real shape than using a sphere), you find that the distance from the equator to 35 degrees of latitude (N or S) is 2230.86 nautical miles.
Total distance from 35N to 35S is twice that at 4461.72 nautical miles (8262.44). A big difference.

If you want to use the real shape of the Earth and follow along the "sea level" of the geoid (meaning the mathematical representation of the real shape of the Earth, minus the mountains), then you would have to determine along which longitude you want to measure.

The lazy way: Measure the distance on Google Earth from some point at 35 N to a point on the same longitude at 35 S. It gives values around 7740 km.

You may want to use more modern coefficients (the nautical mile and the km have been redefined recently to fit with the S.I. system). I'm using values as they were known around 1960.
For example, I find the distance on the ellipsoid (8262.44) to be a bit high. The real distance is probably smaller as the modern value of Earth's "oblateness" is 1/298 -- the planet is a tiny bit less "bulgy" at the equator than what mariners thought in the previous century.

Old definitions:
Earth's circumference = 360 degrees * 60 minutes per degree = 21,600 nautical miles.
Earth's circumference = 400 grades * 100 centigrade per grade = 40,000 km.

21,600 / 40,000 = 0.54 exactly
40,000 / 21,600 = 1.851851851...

Modern values:
The metre was redefined in relation to the speed of light.
Earth's circumference is now set at 40,075,036 m
(40,075 km).
The nautical mile is defined as 1.852 km (exactly)
(circumference = 21,638.8 n.m.). Therefore, the new nautical mile is a tiny bit shorter than the original nautical mile.
The modern km is a very, very tiny bit smaller than the original km.

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I just thought of another way:

Going all around the meridien (360 degrees) is Earth's circumference. Going from 35N to 35S (the short way, I presume) is 70 degrees.
Therefore 70/360 of the whole circumference is
(70/360)*40,000 km = 7,777.7... km.

If you use terrestrial miles (as in the USA) and a value of 25,000 miles for the circumference, then you would get
(70/360)*25,000 = 4,861.111111... miles.