One city is due north of another on the same meridian. Their latitudes are 45 degrees 9 minutes north and 35 degrees 5 minutes north. Taking the radius of the Earth as 3960 miles, find the distance between them.

Step-by-step solution

1. 1

   Recognise this as an arc-length problem. Because both cities share a meridian, the path between them is an arc of a great circle of radius $r=3960$ miles, and

   $s=r\theta,\qquad \theta\ \text{in radians}.$

   The central angle $\theta$ is exactly the difference in latitude.

2. 2

   Subtract the latitudes in degrees and minutes. Borrow one degree ($60'$) so the minutes subtract cleanly:

   $45^\circ 9'-35^\circ 5'=10^\circ 4'.$

3. 3

   Convert the minutes to a decimal degree. One minute is $\tfrac{1}{60}$ of a degree:

   $4'=\frac{4}{60}=0.06\overline{6}^\circ\ \Longrightarrow\ \theta=10.0667^\circ.$

4. 4

   Convert degrees to radians. The formula $s=r\theta$ is only valid in radians:

   $\theta=10.0667\times\frac{\pi}{180}=0.1756986\ \text{rad}.$

   Using $10.0667$ directly would inflate the answer by a factor of about $57$ — the classic slip in arc-length problems.

5. 5

   Multiply by the radius.

   $s=3960\times 0.1756986=695.76\ \text{miles}.$

6. 6

   Check with the degrees-per-mile shortcut. The Earth's circumference is $2\pi(3960)=24{,}881$ miles, so one degree of latitude is $\tfrac{24881}{360}=69.1$ miles. Then $10.0667\times 69.1=695.6$ miles, agreeing to within rounding. The cities are roughly $696$ miles apart.