Chapter 4.8, Problem 23E

To determine

To find: how far apart are the towns.

Answer to Problem 23E

The distance between the two towns is 11.81 kilometers.

Explanation of Solution

Given information:

A passenger in an airplane at an altitude of 10 kilometers sees two towns directly to the east of the plane. The angles of depression to the towns are $28°$ and $55°$ .

  

Calculation:

Consider an airplane at an altitude of 10 kilometers such that the angle of depression to the towns lying directly to the east of the plane is $28°$ and $55°$ .

  

Note that from the above figure,

  $\begin{array}{c}A=90°-55°\\ =35°\end{array}$

Since the tangent of the angle is given by the ratio of the length of the perpendicular to the base

  $\begin{array}{c}\tan A=\frac{d_1}{10}\\ d_1=10\times \tan 35°\\ =10\times 0.7002\\ =7\end{array}$

Also, from the above figure

  $\begin{array}{c}B=\frac{d_2}{10}\\ d_2=10\times \tan 62°\\ =10\times 1.8807\\ =18.81\end{array}$

Therefore, the distance between the two towns is

  $\begin{array}{c}{d}_2-d_1=18.81-7\\ =11.81\end{array}$

Hence, the distance between the two towns is 11.81 kilometers.