Chapter 7, Problem 71RE

(a)

To determine

To express:The viewing angle $\theta$ as a function of the distance $x$ between the driver and the sign.

Answer to Problem 71RE

Theviewing angle $\theta$ is expressed as $\theta ={\tan }^{-1}\left(\frac{10}{x}\right)$ .

Explanation of Solution

Given information:

The given figure is as follows.

  

Calculation:

From the above figure,

  $\begin{array}{c}\tan \theta =\frac{\text{opp}}{\text{adj}}\\ =\frac{10}{x}\\ \theta ={\tan }^{-1}\left(\frac{10}{x}\right)\end{array}$

Therefore, the viewing angle $\theta$ is expressed as $\theta ={\tan }^{-1}\left(\frac{10}{x}\right)$ .

(b)

To determine

To find:The distance $x$ at which the sign first become legible.

Answer to Problem 71RE

The sign first becomes legible at s distance of $286.4\text{ft}$ from the sign.

Explanation of Solution

Given information:

It is given that the sign is legible when the viewing angle is $2°$ or greater.

Calculation:

From the given information,

  $\begin{array}{c}\theta \ge 2\\ {\tan }^{-1}\left(\frac{10}{x}\right)\ge 2\\ \frac{10}{x}\ge \tan 2\\ \frac{10}{x}\ge 0.03492\end{array}$

Simplify further.

  $\begin{array}{c}x\le \frac{10}{0.03492}\\ \le 286.4\text{ft}\end{array}$

Therefore, the sign first becomes legible at s distance of $286.4\text{ft}$ from the sign.