Calculating Distance Between Two Places Along 66° Latitude (Rhumb Line)

[prinsinn]

Homework Statement

Two places, A (66°N and 20°W) and B (66°N and 90°A). Find the distance between them if you go along the 66° latitude (rhumb line).

Homework Equations

The Attempt at a Solution

The answer is 4972 km but I don´t know how to calculate this. Please help me.

 

[Dick]

A constant latitude curve is just a circle. Can you find it's radius? At 0 latitude it's just the radius of the earth. As you approach the poles it goes to zero. Think of a trig function to apply.

 

[prinsinn]

The radius of the Earth is 6370 km.

I know how to calculate the great circle (the shortest distance). This is how I did that.

a = 90-66 = 24
b = 90-66 = 24
C = 20+90 = 110

I put these numbers into the formula
cos(c) = cos(a)cos(b) + sin(a)sin(b)cos(C)

which gives me c = 0.7779 = 0.67935 rad
and 0.67935 * 6360(radius of earth) = 4327 km.

This is how to find the great circle but I have tried so hard to find the distance if I go along the 66° latitute but I don´t get the right answer. Please show me how to do that.

 

[Dick]

Compute the radius of the circle (not a great circle) following 66 degrees latitude. Take a point on that circle, connect it perpendicularly to the Earth's axis and then connect that to the Earth's center. You've just drawn a right triangle. The hypotenuse is the radius of the Earth and the first leg is the radius of the latitude circle. Find an angle in that triangle and use trig to find the radius of the latitude circle.

 

[prinsinn]

Ahh, I'm getting closer,
I think the radius of the 66° latitute is (6370/90)*66 = 4671.

I don't know what to do next. I think I have to find the angle between places A and B and divide it by 360 and multiply by the new radius, 4671.


I don't unerstand how to take a point on the circle, connect it perpendicularly to the Earth's axis and then connect that to the Earth's center.

 

[Dick]

No. The radius of the 66 degree latitude circle 6370*cos(66 degrees). It's really hard to explain how to draw the triangle in words and I see I'm failing so take my word for it and ask someone to draw the picture. Yes, then you find the angle difference, divide by 360 and multiply by r. But don't forget to multiply by 2*pi as well. The circumference of a circle is 2*pi*r.

 

[prinsinn]

Thank you so much, I did it right, you are a genius.

The radius of the 66° latitute is 6370*cos(66) = 2590.9 km
The circumference of 66° latitute is 2590.9*pi*2=16279 km
the angle between places A and B is 110°

so the answer is

(110/360) * 16279 = 4974 km