Chapter 4.7, Problem 103E

a.

To determine

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

Answer to Problem 103E

  $\theta =\arcsin \frac{5}{s}$

Explanation of Solution

Given:

A boat is pulled in by means of a winch located on a dock 5 feet above the deck of the boat.

Let $\theta$ be the angle of elevation from the boat to the winch and let s be the length of the rope from the winch to the boat.

  

Now it is known that,

  $\begin{array}{c}\sin \theta =\frac{opp}{hyp}\\ =\frac{5}{s}\end{array}$

Take inverse of the above function.

  $\theta =\arcsin \frac{5}{s}$

b.

To determine

To write: $\theta$ when $s=40\text{ feet}$ and when $s=20\text{ feet}$ .

Answer to Problem 103E

  $\theta \left(40\right)\approx 0.125$

  $\theta \left(20\right)\approx 0.25$

Explanation of Solution

From part (a),

  $\theta =\arcsin \frac{5}{s}$ ….. (1)

First substitute $s=40$ in equation (1),

  $\begin{array}{c}\theta \left(40\right)=\arcsin \frac{5}{40}\\ =\arcsin \frac{1}{8}\\ \approx 0.125\end{array}$

Now substitute $s=20$ in equation (1),

  $\begin{array}{c}\theta \left(20\right)=\arcsin \frac{5}{20}\\ =\arcsin \frac{1}{4}\\ \approx 0.25\end{array}$