Chapter 12.5, Problem 12P

To determine

Find the height of the tower.

Answer to Problem 12P

49.8 m

Explanation of Solution

Given:

From a point 250 m from the base of a vertical cliff, the angles of elevation to the top and bottom of a radio tower on top of the cliff are $62.2°$ and $59.5°$ .

Formula Used:

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

Calculation:

Represent the given situation

  

Use the tan formula :

In triangle ACD:

  $\begin{array}{l}\tan 62.2°=\frac{\text{length of opposite side to 62}\text{.2}°}{\text{length of adjacent side to 62}\text{.2}°}\\ \Rightarrow \tan 62.2°=\frac{AC}{250}\\ \Rightarrow AC=250\tan 62.2°\mathrm{.........................}[\text{multiply each side by 250}]\\ \Rightarrow AC\approx 474.2\end{array}$

In triangle BCD:

  $\begin{array}{l}\tan \text{59}\text{.5}°=\frac{\text{length of opposite side to 59}\text{.5}°}{\text{length of adjacent side to 59}\text{.5}°}\\ \Rightarrow \tan \text{59}\text{.5}°=\frac{BC}{250}\\ \Rightarrow BC=250\tan \text{59}\text{.5}°\mathrm{.........................}[\text{multiply each side by 250}]\\ \Rightarrow BC\approx 424.4\end{array}$

So, the height of the Radio Tower = AB = AC-BC = 474.2 − 424.4 $\approx 49.8$ m.