Differnece in Latitudes

[math-a-phobic]

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!

 

[soroban]

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?

             * * *
          *         *
        *             *
       *             / *
                   /     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: $\displaystyle \,4.655^o$

 

[royhaas]

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.