Chapter 6.1, Problem 90AYU

Problems 89 and 90 require the following discussion:

The shortest distance between two points on Earth's surface can be determined from the latitude and longitude of the two locations. For example, if location 1 has $\left(lat,\text{lon}\right)=\left({\alpha }_1,{\beta }_1\right)$ and location 2 has $\left(lat,lon\right)=\left({\alpha }_2,{\beta }_2\right)$, the shortest distance between the two locations is approximately $d=r{\cos }^{-1}\left[\left(\cos {\alpha }_1\cos {\beta }_1\cos {\alpha }_2\cos {\beta }_2\right)+\left(\cos {\alpha }_1\sin {\beta }_1\cos {\alpha }_2\sin {\beta }_2\right)+\left(\sin {\alpha }_1\sin {\alpha }_2\right)\right]$, where $r=radiusofEarth\approx 3960$ miles and the inverse cosine function is expressed in radians. Also, $N$ latitude and $E$ longitude are positive angles, and $S$ latitude and $W$ longitude are negative.

Source: www. Infoplease.com

Shortest Distance from Honolulu to Melbourne, Australia Find the shortest distance from Honolulu to Melbourne, Australia, latitude $37°47\text{'}S$, longitude $144°58\text{'}E$. Round your answer to the nearest mile.