Chapter 12.8, Problem 10P

To determine

Find the distance between the campsites.

Answer to Problem 10P

About 294.13 meters.

Explanation of Solution

Formula Used:

Law of Cosine :

  $a^2=b^2+c^2-2bc\cos A$

Law of Sine:

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

  

Given:

From the top of an observation post that is 90 m high , a ranger sights a campsite at an angle of depression of $10°$ .Turning in a different direction, the ranger sees another campsite at an angle of depression of $13°$ .The angle between these two lines of sight is $35°$ .

Calculation:

Represent the situation and solve :

  

Find the line of site of campsite A:

  $\begin{array}{l}\tan \theta =\frac{\text{length of the side opposite to }\theta }{\text{length of the side adjacent to }\theta }\\ \Rightarrow \tan 80°=\frac{a}{90}\\ \Rightarrow a=90\tan 80°\mathrm{.............}[\text{multiply each side by 90]}\\ \Rightarrow a\approx 510.42\text{ m}\end{array}$

  

Find the line of site of campsite B:

  $\begin{array}{l}\tan \theta =\frac{\text{length of the side opposite to }\theta }{\text{length of the side adjacent to }\theta }\\ \Rightarrow \tan 77°=\frac{b}{90}\\ \Rightarrow b=90\tan 77°\mathrm{.............}[\text{multiply each side by 90]}\\ \Rightarrow b\approx 389.83\text{ m}\end{array}$

Now, draw the combined triangle that represents the lines of sights from the ranger to the campsites :

  

We need to find AB:

Use Law of Cosine:

  $\begin{array}{l}A{B}^2={510.42}^2+{389.83}^2-2(510.42)(389.83)\cos 35°\\ \Rightarrow AB=\sqrt{{510.42}^2+{389.83}^2-2(510.42)(389.83)\cos 35°}\\ \Rightarrow AB\approx 294.13\text{ m}\end{array}$

So, the distance between the campsites is about 294.13 meters.