Differnece in Latitudes

math-a-phobic

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Feb 10, 2006
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Problem: Assuming tha tthe earth is a sphere of radius 4000 miles, what is the difference in latitudes of two cities, one of which is 325 miles due north of the other.


I drew a picture and made the radius of the circle equal to 4000 miles and had 2 points. I made one point at 0 miles and the other at 325 miles. After multiplying 4000 by (325/Pi) I received 3.918 degrees. I know the answer is 4.655 degrees, but I don't know how to correctly solve the problem. Please help! Thanks! :)
 
Hello, math-a-phobic!

Assuming tha tthe earth is a sphere of radius 4000 miles,
what is the difference in latitudes of two cities,
one of which is 325 miles due north of the other?
Code:
             * * *
          *         *
        *             *
       *             / *
                   /     325
      *          / θ    *
      *        *--------* 
      *           4000  *

       *               *
        *             *
          *         *
             * * *
We have a central angle \(\displaystyle \theta\); the radius is \(\displaystyle r\,=\,4000\) miles.
The length of arc is \(\displaystyle s\,=\,325\) miles.

Length-of-Arc formula: \(\displaystyle \,s\:=\:r\theta\) . . . where \(\displaystyle \theta\) is in radians.

So we have: \(\displaystyle \,325\:=\:4000\theta\;\;\Rightarrow\;\;\theta\:=\:\frac{325}{4000}\:=\:0.08125\) radians.

And 0.08125 radians equals: \(\displaystyle \,\frac{0.08125\text{ rad}}{1}\,\times\,\frac{180^o}{1\text{ rad}} \;= \;4.655282085^o\)

Therefore, the difference in latitude is: \(\displaystyle \,4.655^o\)
 
Use the formula \(\displaystyle S=\theta R\), where S is the distance and R is the radius, to solve for the angle \(\displaystyle \theta\) in radians. Then convert that to degrees.
 
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