Chapter 12.7, Problem 5P

To determine

To determine the length of shortest distance.

Answer to Problem 5P

The length of shortest distance is 3280m

Explanation of Solution

Given information :

Angles of observation to the fire are given ${52}^{\circ }$ and ${41}^{\circ }$

Distance between stations is 5000m

Here third angle is,

  $\left({180}^{\circ }-{52}^{\circ }-{41}^{\circ }\right)={87}^{\circ }$

We have to find the side opposite to the angle ${41}^{\circ }$ as the shortest distance would be opposite to shortest angle.

Now according to law of sines,

In any triangle ABC,

  

  $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

Hence,

  

  $\begin{array}{l}\frac{5000}{\sin {87}^{\circ }}=\frac{b}{\sin {41}^{\circ }}\\ b=\frac{\sin {41}^{\circ }\times 5000}{\sin {87}^{\circ }}\\ b=3280\end{array}$

Hence the length of shortest side is 3280 in.