Q79. - Precalculus Enhanced with Graphing Utilities

Question

Shortest Distance from Chicago to Honolulu Find the shortest distance from Chicago, latitude 41°50 N, longitude

87°37W to Honolulu, latitude 21°18N, longitude 157°50W. Round your answer to the nearest mile.

Step-by-Step Solution

Answer

The shortest distance between two locations on the earth's surface is 4250 miles.

Step 1. Given Information.

Coordinates of two locations on the earth's surface:

1st Chicago:    $\left(lat,lon\right)=\left(41°50N,87°37W\right)$

2nd  Honolulu:$\left(lat,lon\right)=\left(21°18N,157°50W\right)$

We need to determine the shortest distance between the given locations.

The shortest distance can be calculated by using the formula,

$d=r{\cos }^{-1}\left[\left(\cos {\alpha }_1\cos {\beta }_1\cos {\alpha }_2\cos {\beta }_2\right)+\left(\cos {\alpha }_1\sin {\beta }_1\cos {\alpha }_2\sin {\beta }_2\right)+\left(\sin {\alpha }_1\sin {\alpha }_2\right)\right]$

Step 2. Convert given coordinates into degrees

For Chicago:

$$\begin{gathered} {\alpha }_1=41°50\text{'}=\left(41+\frac{50}{60}\right)=41.83 \\ {\beta }_1=87°37\text{'}=-\left(87+\frac{37}{60}\right)=-87.61 \end{gathered}$$

For Honolulu:

$$\begin{gathered} {\alpha }_2=21°18\text{'}N=21.3 \\ {\beta }_2=157°50\text{'}W=-157.8 \end{gathered}$$

Step3. Finding distance.

The substitute the values and calculate,

$\begin{array}{rcl}d& =& r{\cos }^{-1}\left[\left(\cos {\alpha }_1\cos {\beta }_1\cos {\alpha }_2\cos {\beta }_2\right)+\left(\cos {\alpha }_1\sin {\beta }_1\cos {\alpha }_2\sin {\beta }_2\right)+\left(\sin {\alpha }_1\sin {\alpha }_2\right)\right]\\ d& =& r{\cos }^{-1}\left[\left(\cos \left(41.83\right)\cos \left(-87.61\right)\cos \left(21.30\right)\cos \left(-157.83\right)\right)+\left(\cos \left(41.83\right)\sin \left(-87.61\right)\cos \left(21.30\right)\sin \left(-157.83\right)\right)+\left(\sin \left(41.83\right)\sin \left(21.30\right)\right)\right]\\ & =& 3960{\cos }^{-1}\left[\left(-0.02673\right)+\left(0.261\right)+\left(0.242\right)\right]\\ & =& 3960\left(61.49·\frac{\mathrm{\pi }}{180}\right)\\ & =& 4250.22\end{array}$