Chapter 4.8, Problem 29E

To determine

Tofind:the angle of depression from the top of the tower to a cell phone user.

Answer to Problem 29E

The angle of depression from the top of the tower to the cell phone user is

  ${\text{tan}}^{-1}\left(5.11\times {10}^{-2}\right)=2.06^{″}$

Explanation of Solution

Given:

A cellular telephone tower that is 150 feet tall

Top of mountain 1200 feet above sea level.

Concept used:

Apply the trigonometric identity:

  $\text{tanθ=}\frac{\text{perpendicular}}{\text{base}}$

Calculation:

Let consider the height of the telephone tower, mountain and the cell phone user to be

  $\text{AF,EF and DC}\left(\text{orBE}\right)$ respectively.

Therefore, the figure can be drawn as the right-angled triangle shown below.

In the figure drawn above $\text{θ}$ is the angle of depression from the top of the tower $\left(\text{A}\right)$ to the cell phone user $\left(\text{C}\right)$ .

Therefore, it is also the angle of elevation from C to A.

Use the fact that

  $\text{ED=5miles}$

  $\Rightarrow 26394.357\text{ft}$

  $\text{FE=1200ft}$

  $\text{AF=150ft}$

So, here

  $\text{AB=AF+FE-BE}$

  $\Rightarrow \text{150+1200-400}$

  $\text{AB=950ft}$

Shall now apply the trigonometric identity in $\Delta \text{ABC}$

  $\text{tanθ=}\frac{\text{AB}}{\text{BC}}$

  $\Rightarrow \frac{\text{AF+FB}}{\text{BC}}$

  $\Rightarrow \frac{150+1200}{26394.357}$

  $\Rightarrow 0.0511473$

  $\text{θ=Arctan}\left(5.11\times {10}^{-2}\right)$

Hence the angle of depression from the top of the tower to the cell phone user is

  ${\text{tan}}^{-1}\left(5.11\times {10}^{-2}\right)=2.06^{″}$