Chapter 12.5, Problem 15P

To determine

Find the height of the helicopter.

Answer to Problem 15P

About 659.9 m.

Explanation of Solution

Given:

Two observers 1600 m apart on a straight , flat road measure the angles of elevation of a helicopter hovering over the road between them.The angles are $32°$ and $50.5°$ .

Formula Used:

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

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

Calculation:

Represent the given situation:

  

We need to find h .

We now write equations :

  $\begin{array}{l}\cot 32°=\frac{\text{length of adjacent side to 32}°}{\text{length of opposite side to 32}°}\\ \Rightarrow \cot 32°=\frac{x}{h}\\ \Rightarrow x=h\cot 32°\mathrm{...................}[\text{multiply each side by h]}\end{array}$

  $\begin{array}{l}\tan 50.5°=\frac{\text{length of opposite side to }\theta }{\text{length of adjacent side to }\theta }\\ \Rightarrow \tan 50.5°=\frac{h}{1600-x}\\ \Rightarrow h=(1600-x)\tan 50.5°\mathrm{...................}[\text{multiply each side by 1600-x]}\\ \Rightarrow h=(1600-h\cot 32°)\tan 50.5°\mathrm{.......}[x=h\cot 32°]\\ \Rightarrow h=1600\tan 50.5°-h\cot 32°\tan 50.5°\\ \Rightarrow h+h\cot 32°\tan 50.5°=1600\tan 50.5°\mathrm{....}\text{[add }h\cot 32°\tan 50.5°\text{ each side]}\\ \Rightarrow h(1+\cot 32°\tan 50.5°)=1600\tan 50.5°\\ \Rightarrow h=\frac{1600\tan 50.5°}{1+\cot 32°\tan 50.5°}\mathrm{..........................}[\text{divide each side by }1+\cot 32°\tan 50.5°]\\ \Rightarrow h\approx 659.9\end{array}$

So, height of the helicopter is about 659.9 m.