Question

Acapulco, Mexico and Hyderabad, India both lie at 17° north latitude, and lie very nearly halfway around the world from each other in an east-west direction. The radius of Earth at a latitude of 17° is about 3790 miles. Suppose that you could fly from Acapulco directly west to Hyderabad or fly directly north to Hyderabad. Which way would be shorter, and by how much? Use 3960 miles for Earth’s radius. (Hint: To fly directly north, you would go from 17° north latitude to 90° north latitude, and then back down to 17° north latitude.)

Answer 1

In solving this problem we can consider Earth to be a sphere. When we have a circle, then we can use this formula to find arc length:
$L=2 \pi R \frac{C}{360}$
Where:
L= arc length (in this problem it is disance we need to travel)
R = radius of circle (in this problem it is equal to a radius of Earth)
C = angle we need to pass

We are told that two cities lie halfway around the world. If we fly to west this means angle is 180°.
This gives an arc length of:
$L=2* \pi *3960* \frac{180}{360} \\ \\ L=12440.71 miles$

If we want to fly to north we need to go to 90° northern latidtude and then back to 17° latitude. This means angle is:
C=2*(90-17)=2*73°=146°
This gives an arc length of:
$L=2* \pi *3960* \frac{146}{360} \\ \\ L=10090.8 miles$

We can see that flying north is shorter. It is shorter by:
12440.71 miles - 10090.8 miles = 2349.91 miles