Chapter 12.5, Problem 16P

To determine

Find the height of the balloon and its horizontal distance from the nearer end of the field.

Answer to Problem 16P

Height is about 1511 ft and distance from nearer end of the field is about 2182.17 ft.

Explanation of Solution

Given:

The pilot of a hot air balloon sees a field straight ahead that he knows is 1000 ft long.The angles of depression to the ends of the field are $25.4°$ and $34.7°$ .

Formula Used:

  $\tan \theta =\frac{\text{length of opposite side to }\theta }{\text{length of adjacent side to }\theta }$

Calculation:

Represent the given situation:

  

Since the dotted line and the ground are parallel , we can find the angles of the triangles by alternate interior angles .

We need to find h and x .

We now write equations :

  $\begin{array}{l}\tan 34.7°=\frac{\text{length of opposite side to }34.7°}{\text{length of adjacent side to }34.7°}\\ \Rightarrow \tan 34.7°=\frac{h}{x}\\ \Rightarrow h=x\tan 34.7°\mathrm{...............}[\text{multiply each side by x]}\end{array}$

  $\begin{array}{l}\tan 25.4°=\frac{\text{length of opposite side to }25.4°}{\text{length of adjacent side to }25.4°}\\ \Rightarrow \tan 25.4°=\frac{h}{1000+x}\\ \Rightarrow h=(1000+x)\tan 25.4°\mathrm{...............}[\text{multiply each side by (1000+x)]}\end{array}$

So, equate the value of h from both the equations above:

  $\begin{array}{l}x\tan 34.7°=(1000+x)\tan 25.4°\\ \Rightarrow x\tan 34.7°=1000\tan 25.4°+x\tan 25.4°\\ \Rightarrow x\tan 34.7°-x\tan 25.4°=1000\tan 25.4°\mathrm{..........}[\text{subtract }\tan 25.4°\text{ from each side]}\\ \Rightarrow x(\tan 34.7°-\tan 25.4°)=1000\tan 25.4°\\ \Rightarrow x=\frac{1000\tan 25.4°}{\tan 34.7°-\tan 25.4°}\mathrm{.................................}[\text{divide each side by }\tan 34.7°-\tan 25.4°]\\ \Rightarrow x\approx 2182.17\\ \Rightarrow h=2182.17\tan 34.7°\approx 1511\end{array}$

So, height is about 1511 ft and distance from nearer end of the field is about 2182.17 ft.