Chapter 4.7, Problem 108E

(a)

To determine

To write: $\theta$ as a function of $x$ .

Answer to Problem 108E

  $\theta ={\tan }^{-1}\left(\frac{6}{x}\right)$

Explanation of Solution

Given information:

An airplane flies at an altitude of 6 miles toward a point directly over an observer. Consider $\theta$ and $x$ as shown in the figure.

Calculation:

Consider the figure

It is known that

  $\tan \theta =\frac{\text{opp}}{\text{adj}}$

In the above figure,

  $\begin{array}{c}\tan \theta =\frac{6}{750}\\ \theta ={\tan }^{-1}\left(\frac{6}{x}\right)\end{array}$

Therefore $\theta$ in function of $x$ is written as ${\tan }^{-1}\left(\frac{6}{x}\right)$ .

(b)

To determine

To find $\theta$ , when $x=12$ miles, and $x=7$ miles.

Answer to Problem 108E

The value of $\theta$ is $40.6°$ and $80.54°$ when $x$ is 7 miles and 1 miles respectively.

Explanation of Solution

Calculation:

Consider $x=7$

  $\begin{array}{c}\theta ={\tan }^{-1}\left(\frac{6}{7}\right)\\ =40.6\end{array}$

Consider $x=1$

  $\begin{array}{c}\theta ={\tan }^{-1}\left(\frac{6}{1}\right)\\ ={\tan }^{-1}\left(6\right)\\ =80.54\end{array}$

Therefore, the value of $\theta$ is $40.6°$ and $80.54°$ when $x$ is 7 miles and 1 miles respectively.