Chapter 12.5, Problem 11P

To determine

Find the distance between the fires.

Answer to Problem 11P

358.4 ft.

Explanation of Solution

Given:

From the top of a 135 ft observation tower, a park ranger sights two forest fires on opposite sides of the tower.If their angles of depression are $42.5°$ and $32.6°$ .

Formula Used:

  $\tan \theta =\frac{\text{length of opposite side to }\theta }{\text{length of adjacent side to }\theta }$

Calculation:

Represent the given situation

  

Since the dotted line and the base of both the triangles are parallel , so, by alternate interior angles , we can find on angle of each triangle.

We need to find x + y .

Use the tan formula :

  $\begin{array}{l}\tan 42.5°=\frac{\text{length of opposite side to 42}\text{.5}°}{\text{length of adjacent side to 42}\text{.5}°}\\ \Rightarrow \tan 42.5°=\frac{135}{x}\\ \Rightarrow x=\frac{135}{\tan 42.5°}\mathrm{.........................}[\text{multiply each side by }\frac{x}{\tan 42.5°}]\\ \Rightarrow x\approx 147.327\end{array}$

  $\begin{array}{l}\tan 32.6°=\frac{\text{length of opposite side to 32}\text{.6}°}{\text{length of adjacent side to 32}\text{.6}°}\\ \Rightarrow \tan 32.6°=\frac{135}{y}\\ \Rightarrow y=\frac{135}{\tan 32.6°}\mathrm{.........................}[\text{multiply each side by }\frac{y}{\tan 32.6°}]\\ \Rightarrow y\approx 211.094\end{array}$

So, the distance between the fires is x + y = 147.327 + 211.094 = about 358.4 ft.