Quito a city in ecuador and kampala a city in uganda both lie on the equator. The longitude of quito is 82 degree 30 minutes west and that of kampala is 37 degree 30 minutes east.what is the distance from quito to kampala (a) along the shortest path (b) along a direct (through the earth)path? The radius of earth is 6.4*10^8 metre.

Question

bharat bajaj · Answer

Both the points lie on the equator, so forget about the earth. Think of just a equatorial circle.
a) Say A is Quito and B is Kampala.
Angle subtended by A at the centre of circle = - 82 degrees 30 minutes.
Angle subtended by B at the centre of circle = + 37 degrees 30 minutes
Total angle subtended at centre = 120 degrees

Hence, shortest path would be along the equator = Length of arc = 120/360 x Circumference = 2.pi.r/3
b) Direct path would be the line joing the two points. We have to find the length of the chord which subtends 120 degrees at the centre.
Make a line from centre to chord. Say this point is D on AB. It is the midpoint. So, DB subtends 60 degrees at the centre.
DB = r sin 60
Length of chord = 2r sin 60
Thanks

Bharat Bajaj
IIT Delhi

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