distances using longitude and latitude

Question

there is a site in Bowling Green, KY, with the latitude 36.98 degrees N and longitude 86.5 degrees W. measure the distance from this place to that site in Hiawatha National Forest, in the Upper Peninsula of Michigan, with the same longitude and latitude 46.21 degrees N. Considering the radius of the earth to be 6371km, find the distance between the places on a straight path, in km.

Answer 1

Oh Shucks, I was looking forward to a good Spherical Trigonometry / Navigation problem ! This one is straight along a Meridian.

Distance from Bowling Green, KY (Lat 36.98 degrees N, Long 86.5 degrees W) to Hiawatha National Forest, in the Upper Peninsula of Michigan (Lat 46.21 degrees N, same Longitude), so straight Meridian distance.

Average Earth radius = 6371km is used as a standard value [ between Equatorial radius=6,378.1370 km and Polar radius=6,356.7523 km ]

Arc distance along a circle: s = rθ

2 approaches:

a. From Earth's radius: s = rθ

θ = (46.21º - 36.98º) = 9.23º... we need Radians here: s = rθ = (6371km)(9.23º)(π/180º) = 1026.32 km

b. From ΔLat to Nautical miles (1 NM = 1 minute of arc on Earth's surface), etc...

θ = (46.21º - 36.98º) = 9.23º ==> (9.23º)(60 minutes of arc / 1º)(1 Nautical Mile / 1')(1.151 Statute Mile / Nautical Mile)(1.61 km / 1 mile)

= 9.23*60*1.151*1.61 miles = 1026.25 km