Measuring Distances Using Lat / Long Coordinates (Math Problem Sample)

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Measuring Distances Using Lat/Long Coordinates
Part 1
Considering two landmark buildings, Andre in Tempe and Fort Lowell Museum in Tucson, from the google map, their coordinates approximated to the nearest 4 decimal places yield Andre (35.07850S, 150.68750E) and Fort Lowell Museum (32.22170N, 110.92640W).
Distance in Miles Using Pythagoras’ Theorem. Acknowledging that to be able to use Pythagoras’ Theorem, the obligate spheroid shape of the Earth will be assumed to be a plane surface. Hence, the sketch below would denote the two locations.
Now, the distances of the sides marked x and y must be obtained.
Normally, 10 = 180π ≈ 57.30 to the nearest four significant figures. Thus, converting both the latitudes and longitudes into radius,
Andre (35.07850S, 150.68750E) translates to (-0.6122, 2.6298) and Fort Lowell Museum (32.22170N, 110.92640W) translates to (0.5623, -1.9359)
Then:
Distance x = [ Cos (lat)× (longitudinal Difference)]
= [Cos (0.5623)× (1.9359 + 2.6298) × 3963 miles]
= 18092.99776 miles.
Similarly,
Distance y = [3963 (Latitudinal Difference)]
= [3963 (0.6122 + 0.5623) miles]
= 4654.5435 miles.
Recall that the differences were obtained by addition since they are on opposite sides of the GMT and Equator respectively.
Hence, from Pythagoras Theorem, the distance between the two buildings is given as:
Distance = (x2+y2) = (18092.99776 2+4654.5435 2)
= 18 681.9783 miles.
Distance in Miles Using Spherical Geometry. Now, assume that the shape of the earth is a perfect sphere. Then, the following sketch can be formulated.
Consider the spherical triangle above, using spherical geometry and the Cosine Rule,
Cos g = {Sin lat1 × Sin lat2} + Cos lat1 × Cos lat2 × Cos (long2 – long1)
= Sin (-0.6122) × Sin (0.5623) + Cos (-0.6122) × Cos (0.5623) Cos (4.5657)
= 0.9966
Therefore, given that the average radius (R) of the Earth is 3963 miles, then, the distance between the two buildings would be given as:
= R × arcCos g.
= 3963 × arcCos (0.9966) miles. (arcCos denotes the Cos-1)
= 18 169.90144 miles.
Percentage Difference between the Two Measurements Using. The percentage difference would be given using the relationship below.
That is, % Difference = 18 681.9783 – 18 169.90144 18 169.90144 ×100
= 2.818 %
Part 2
Now, consider the case of New York City and Los Angeles. Using the same procedure, it is possible to obtain their coordinates: New York City (40.7128° N, 74.0059° W) and Los Angeles (34.0522° N, 118.2437° W).
Converting the degre...