Chapter 7, Problem 72RE

(a)

To determine

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

Answer to Problem 72RE

Theviewing angle $\theta$ is expressed as $\theta ={\tan }^{-1}\left(\frac{420}{x}\right)-{\tan }^{-1}\left(\frac{380}{x}\right)$ .

Explanation of Solution

Given information:

The given figure is as follows.

  

Calculation:

Let $\alpha$ and $\beta$ be the angles of elevations to the top of the tower and building respectively.

From the above figure, in the $\Delta ABD$

  $\begin{array}{c}\tan \beta =\frac{\text{opp}}{\text{adj}}\\ =\frac{380}{x}\\ \beta ={\tan }^{-1}\left(\frac{380}{x}\right)\end{array}$

From the above figure, in the $\Delta ABC$

  $\begin{array}{c}\tan \alpha =\frac{\text{opp}}{\text{adj}}\\ =\frac{420}{x}\\ \alpha ={\tan }^{-1}\left(\frac{420}{x}\right)\end{array}$

From the figure, the viewing angle $\theta$ is $\theta =\alpha -\beta$ .

  $\begin{array}{c}\theta =\alpha -\beta \\ ={\tan }^{-1}\left(\frac{420}{x}\right)-{\tan }^{-1}\left(\frac{380}{x}\right)\end{array}$

Therefore, the viewing angle $\theta$ is expressed as $\theta ={\tan }^{-1}\left(\frac{420}{x}\right)-{\tan }^{-1}\left(\frac{380}{x}\right)$ .

(b)

To determine

To find:The distance from the building at which the viewing angle is large as possible.

Answer to Problem 72RE

At a distance of $399.5\text{ft}$ , viewing angle is maximum.

Explanation of Solution

Calculation:

By graphing the equation, the value of $x$ can be found which makes $\theta$ as large as possible.

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

  

Therefore, at a distance of $399.5\text{ft}$ , viewing angle is maximum.