Two cities lie on the same meridian, one at latitude 45 degrees 9 minutes north and the other at 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

   Turn the geography into a circle problem. Latitude is the central angle measured from the equator, so two places on the same meridian sit on one great circle of radius $3960$ miles. The angle at the centre between them is simply the difference of their latitudes - no spherical trigonometry is needed precisely because they share a meridian.

2. 2

   Subtract in degrees and minutes. Since $9'>5'$ no borrowing is required:

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

3. 3

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

   $10^\circ 4'=10+\frac{4}{60}=10.0667^\circ.$

4. 4

   Convert to radians - this step is mandatory. The arc length formula $s=r\theta$ is only valid with $\theta$ in radians, because a radian is defined as the angle whose arc equals the radius:

   $\theta=10.0667^\circ\times\frac{\pi}{180}=0.175696\ \text{rad}.$

5. 5

   Apply $s=r\theta$.

   $s=3960\times 0.175696=695.76\ \text{miles}\approx 696\ \text{miles}.$

6. 6

   Check that the size is plausible. One degree of latitude is about $\tfrac{2\pi(3960)}{360}\approx 69.1$ miles, and $10.07\times 69.1\approx 696$, so the answer is consistent.