Chapter 6, Problem 3PS

a.

To determine

The visual representation of the two ranger stations.

Answer to Problem 3PS

  

Explanation of Solution

Given information:

A group of hikers is lost in a national park. Two ranger stations receive an emergency SOS signal from the hikers. Station B is $75$ miles due east of station A. The bearing from station A to the signal is S ${60}^{o}E$ and the bearing from station B to the signal is S ${75}^{o}W$ .

  $(a)$ Draw a diagram that gives a visual representation of the problem.

Calculation:

From the given information, two ranger stations receive an emergency SOS signal from the hikers. Station B is $75$ miles due east of station A. The bearing from station A to the signal is S ${60}^{o}E$ and the bearing from station B to the signal is S ${75}^{o}W$ .

The diagram shows the visual representation of the problem as follows:

  

  $a=\frac{c\sin A}{\sin C},b=\frac{c\sin B}{\sin C}$

b.

To determine

The distance from each station to the SOS signal.

Answer to Problem 3PS

Station B $\approx 53.03miles$ and Station A $\approx 27.45miles$.

Explanation of Solution

Given information:

A group of hikers is lost in a national park. Two ranger stations receive an emergency SOS signal from the hikers. Station B is $75$ miles due east of station A. The bearing from station A to the signal is S ${60}^{o}E$ and the bearing from station B to the signal is S ${75}^{o}W$ .

  $(b)$ Find the distance from each station to the SOS signal.

Calculation:

From the given information, station A to the signal is S ${60}^{o}E$ and the bearing from station B to the signal is S ${75}^{o}W$ .

  $\begin{array}{l}a=\frac{75\sin {30}^o}{\sin {135}^o}\approx 53.03miles\\ b=\frac{75\sin {15}^o}{\sin {135}^o}\approx 27.45miles\end{array}$

Thus, the distance from station B is $\approx 53.03miles$ and the distance from station A is $\approx 27.45miles$

c.

To determine

The distance and the bearing rescue party which is travel to reach the lost hikers.

Answer to Problem 3PS

  $11.03miles$ and $S{21.68}^{o}E$ .

Explanation of Solution

Given information:

A group of hikers is lost in a national park. Two ranger stations receive an emergency SOS signal from the hikers. Station B is $75$ miles due east of station A. The bearing from station A to the signal is S ${60}^{o}E$ and the bearing from station B to the signal is S ${75}^{o}W$ .

  $(c)$ A rescue party is in the park $20$ miles from Station A at a bearing of S ${80}^{o}E$ . Find the distance and the bearing the rescue party must travel to reach the lost hikers.

Calculation:

From the given information, have the difference of $27.45@60$ and $20@80$ .

  $\begin{array}{l}d=\sqrt{{27.45}^2+{20}^2-2(20)(27.45)\cos {20}^o}\approx 11.03miles\\ b=\frac{27.45\sin {60}^o-20\sin {80}^o}{27.45\cos {60}^o-20\cos {80}^o}\approx S{21.68}^{o}E\end{array}$

  

Thus,the distance and the bearing rescue party is $11.03miles$ and $S{21.68}^{o}E$ .