Chapter 9.4, Problem 38PS

Solution

$\left(a\right)$

To find: The angle ${\theta }_2$ .

  ${\theta }_2={44.821}^{\circ }$

Given information:

If a person is in shallow water and look at an object below the surface of the water, the object will look farther away from the person than it really is. This is because when light rays pass between air and water, the water refracts or bend, the light rays. The index of refraction for water is $1.333$ . This is the ratio of the sine of ${\theta }_1$ to the sine of ${\theta }_2$ for the angles ${\theta }_1$ and ${\theta }_2$ as shown below:

  

A person is in $\text{4}\text{feet}$ of water in the swallow end of the pool. He looks down at some goggles at an angle ${\theta }_1={70}^{\circ }$ (measured from the line perpendicular to the surface of water). Find ${\theta }_2$ .

Explanation:

The vertically opposite angle equals to ${\theta }_1$ is marked in the diagram below:

  

It is given that

  $\begin{array}{c}\frac{\sin {\theta }_1}{\sin {\theta }_2}=1.333\\ \frac{\sin \text{}{70}^{\circ }}{\sin {\theta }_2}=1.333\\ \sin {\theta }_2=\frac{\sin \text{}{70}^{\circ }}{1.333}\\ \sin {\theta }_2=\frac{0.9397}{1.333}\\ \sin {\theta }_2=0.7049\\ {\theta }_2={44.821}^{\circ }\end{array}$

$\left(b\right)$

To find: The distances $x$ and $y$ .

  $\begin{array}{l}x=\mathrm{3.9748.}\\ y=10.99\end{array}$

Given information:

Find the distances $x$ and $y$ .

Explanation:

Use the trigonometric ratios in the right triangle containing angle ${\theta }_2$ :

  $\begin{array}{c}\tan {\theta }_2=\frac{x}{4}\\ \tan 44\cdot {821}^{\circ }=\frac{x}{4}\\ 0\cdot 9937=\frac{x}{4}\\ x=4\times 0\cdot 9937\\ x=3\cdot 9748\end{array}$

Use the trigonometric ratios in the right triangle containing angle ${\theta }_1$ :

  $\begin{array}{l}\tan {\theta }_1=\frac{y}{4}\\ \tan {70}^{\circ }=\frac{y}{4}\\ 2\cdot 7475=\frac{y}{4}\\ \Rightarrow y=4\times 2\cdot 7475=10\cdot 99\end{array}$

$\left(c\right)$

To find: The distance $d$ between where the goggles are and where they appear to be.

  $\text{d=7}\cdot \text{0152}\text{feet}\text{.}$

Given information:

The distance $d$ between where the goggles are and where they appear to be.

Explanation:

The distance $d$ is calculated as follows:

  $\begin{array}{l}d=y-x\\ =10\cdot 99-3\cdot 9748\\ =7\cdot 0152\end{array}$

$\left(d\right)$

To find: What happens to distance $d$ as the person moves closer to the goggles.

The distance $d$ increases as the person moves closer to the goggles.

Given information:

Explain what happens to $d$ as the person moves closer to the goggles.

Explanation:

Since the angle ${\theta }_1$ increases as the person moves toward the goggles, it follows that:

The distance $d$ increases with the movement of the person toward the goggles.