Question

If you walked north along one line of longitude until you had changed the latitude by 1 minute of arc (there are 60 minutes per degree), how far would you have walked (in miles)? (This distance is called a nautical mile.)

Short answer

The distance covered from the initial point will be 1.15 miles.

Step-by-step solution

Step 1. Definition of a nautical mile 

A nautical mile is a unit of distance. One nautical mile is equal to the arc length corresponding to 1 minute of latitude.

Step 2. Conversion factors and assumptions

As mentioned in the question, $1\min =\left(\frac{1}{60}\right)°$.

In the conversion from degrees to radians, $1°=\frac{\pi }{180}\mathrm{rad}$.

Take the radius of the earth to be r= 3958.8 miles. Let the arc length be L.

Step 3. Calculating angle subtended by the arc at the center of the earth

Let the angle subtended by the arc formed at the center of the earth be $\theta$. Then, by the conversion factor from minutes to degrees, the value of $\theta$, when the latitude is changed by 1 minute of arc, is$\theta =\left(\frac{1}{60}\right)°$.

Step 4. Converting degrees to radians

Using the conversion factor from degrees to radians, you can write the value of $\theta$as

$\begin{array}{c}\theta =\frac{1}{60}\times \frac{\pi }{180}\\ =\frac{\pi }{10800}\mathrm{rad}\end{array}$

Step 5. Calculating arc length

By using the definition of radian, the arc length can be written as

$\begin{array}{c}L=r\theta \\ =3958.8\times \frac{\pi }{10800}\\ =1.15\mathrm{miles}\end{array}$

Thus, the arc length will be approximately equal to 1.15 miles, which is equal to one nautical mile.