Chapter 4, Problem 95RE

To determine

The distance from the city A to C and bearing from the city A to C.

Answer to Problem 95RE

The distance is 1221mi and the bearing is 85.6^o.

Explanation of Solution

Given:

From city A to B, plane flies 650 miles at the bearing of the 48^o. From the city B to C, the plane flies 810 miles at the bearing of the 115^o.

Draw the diagram to show the path:

Using the alternate and the adjacent angle property:

  $\begin{array}{c}\angle B={48}^0+\left({180}^0-{115}^0\right)\\ ={48}^0+{65}^0\\ ={113}^0\end{array}$

Using the cosine rue, the value of the distance is given as:

  $\begin{array}{c}{x}^2={\left(AB\right)}^2+{\left(BC\right)}^2-2\left(AB\right)\left(BC\right)\cos B\\ ={810}^2+{650}^2-2\left(810\right)\left(650\right)\cos {113}^o\\ =1220.67\\ =1221\text{mi}\end{array}$

Now, evaluating the angle A :

  $\begin{array}{c}\frac{a}{\sin A}=\frac{x}{\sin B}\\ \sin A=\frac{810\sin {113}^o}{1221}\\ =0.61\\ A={37.64}^o\end{array}$

Therefore,

  $\begin{array}{c}\theta ={48}^o+{37.64}^o\\ ={85.6}^o\end{array}$