Chapter 9.5, Problem 46E

a.

To determine

To write an equation that gives the distance d (in feet) your friends is from the top of the cliff as function of the angle of elevation $\theta$ .

Answer to Problem 46E

  $d=200-300\tan \theta$

Explanation of Solution

Given:

You are standing $300$ feet from the base of a $200$ -foot cliff.

Calculation:

The situation can be diagrammed as follows-

  

Now,

  $\begin{array}{l}\tan \theta =\frac{200-d}{300}\\ 300\tan \theta =200-d\\ d=200-300\tan \theta \end{array}$

b.

To determine

To graph the function found in part (a).

Explanation of Solution

Given:

You are standing $300$ feet from the base of a $200$ -foot cliff.

Calculation:

The following is the graph of the function $d=200-300\tan \theta$

  

c.

To determine

To calculate the angle of elevation when your friend has rappelled halfway down the cliff.

Answer to Problem 46E

  $\theta ={18.4}^{\circ }$

Explanation of Solution

Given:

You are standing $300$ feet from the base of a $200$ -foot cliff.

Calculation:

If friend has rappelled halfway down the cliff $d=\frac{200}{2}=100$

From part (a), it is known that

  $\begin{array}{l}d=200-300\tan \theta \\ \Rightarrow 100=200-300\tan \theta \\ \Rightarrow \tan \theta =\frac{1}{3}\\ \Rightarrow \theta ={\tan }^{-1}\frac{1}{3}\\ \Rightarrow \theta ={18.4}^{\circ }\end{array}$