Chapter 12.8, Problem 4P

To determine

Find the height of the mast.

Answer to Problem 4P

About 388 ft.

Explanation of Solution

Given:

A vertical television mast is mounted on the roof of a building.From a point 750 ft from the base of the building , the angles of elevation to the bottom and top of the mast measure $34°$ and $50°$ respectively.

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:

  $\begin{array}{l}\tan \text{50}°=\frac{\text{length of opposite side to 50}°}{\text{length of adjacent side to 50}°}\\ \Rightarrow \tan \text{50}°=\frac{AC}{750}\\ \Rightarrow AC=750\tan \text{50}°\mathrm{.........................}[\text{multiply each side by 750}]\\ \Rightarrow AC\approx 893.8\end{array}$

In triangle BCD:

  $\begin{array}{l}\tan \text{34}°=\frac{\text{length of opposite side to 34}°}{\text{length of adjacent side to 34}°}\\ \Rightarrow \tan \text{34}°=\frac{BC}{750}\\ \Rightarrow BC=750\tan \text{34}°\mathrm{.........................}[\text{multiply each side by 750}]\\ \Rightarrow BC\approx 505.9\end{array}$

So, the height of the Mast = AB = AC-BC = 893.8 − 505.9 $\approx 387.9\approx 388$ ft.