Chapter 4.7, Problem 55E

Solution

(a)

To write : $\theta$ as a function of the height $s$ of the balloon.

The required function is $\theta ={\tan }^{-1}\frac{s}{500}$ .

Given information :

The hot-air balloon festival held each year in Phoenix, Arizona, is a popular event of photographers. Jo Silver, an award-winning photographer at the event, watches a balloon rising from the ground level from a $500\text{ ft}$ away on level ground.

  

Explanation :

Consider the given figure.

In the given triangle,

  $\begin{array}{c}\tan \theta =\frac{s}{500}\\ \theta ={\tan }^{-1}\frac{s}{500}\end{array}$

Therefore, the required function is $\theta ={\tan }^{-1}\frac{s}{500}$ .

(b)

To calculate : The change in is greater as $s$ changes from $10\text{ ft to }20\text{ ft}$ or as s changes from $200\text{ ft to }210\text{ ft}$

The change in $\theta$ is more when $s$ changes from $10\text{ ft to }20\text{ ft}$ than when $s$ changes from $200\text{ ft to }210\text{ ft}$ .

Given information :

The hot-air balloon festival held each year in Phoenix, Arizona, is a popular event of photographers. Jo Silver, an award-winning photographer at the event, watches a balloon rising from the ground level from a $500\text{ ft}$ away on level ground.

  

Calculation :

Consider the function $\theta ={\tan }^{-1}\frac{s}{500}$ .

When $s$ changes from $10\text{ ft to }20\text{ ft}$ then $\theta$ will be,

  $\begin{array}{c}\theta ={\tan }^{-1}\left(\frac{20}{500}\right)-{\tan }^{-1}\left(\frac{10}{500}\right)\\ ={\tan }^{-1}\left(0.04\right)-{\tan }^{-1}\left(0.02\right)\\ =2.2906-1.1458\\ =1.1448°\end{array}$

When $s$ changes from $200\text{ ft to }210\text{ ft}$ then $\theta$ will be,

  $\begin{array}{c}\theta ={\tan }^{-1}\left(\frac{210}{500}\right)-{\tan }^{-1}\left(\frac{200}{500}\right)\\ ={\tan }^{-1}\left(0.42\right)-{\tan }^{-1}\left(0.4\right)\\ =22.78-21.80\\ =0.98°\end{array}$

Therefore, the change in $\theta$ is more when $s$ changes from $10\text{ ft to }20\text{ ft}$ than when $s$ changes from $200\text{ ft to }210\text{ ft}$ .

(c)

To explain : The axis representing height and the axis representing angle $\theta$ .

The $x-$ axis represents the angle $\theta$ and the $y-$ axis represents the height.

Given information :

The hot-air balloon festival held each year in Phoenix, Arizona, is a popular event of photographers. Jo Silver, an award-winning photographer at the event, watches a balloon rising from the ground level from a $500\text{ ft}$ away on level ground.

  

Explanation :

Here, the $x-$ axis represents the angle $\theta$ and the $y-$ axis represents the height. The angle cannot be of more than $90°$ .