Chapter 4, Problem 95RE

To determine

Find the distance between the two cities and the bearing from city $A$ to city $C$ .

Answer to Problem 95RE

  $\boxed{x\approx 1221miles}$ , $\boxed{\theta =85.6°}$

Explanation of Solution

Given information:

From city A to city B, a plane flies $650$ miles at a bearing of $48°$ . From city $B$ to city $C$ , the plane flies $810$ miles at a bearing of $115°$ . Find the distance from city $A$ to city $C$ and the bearing from city $A$ to city $C$ .

Calculation:

Here, we will first make the diagram of the given problem.

Now, we have to find $\theta$ and $x$ .

We will use alternate and adjacent angle property,

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

Now, by using the cosine rule:

  $\begin{array}{l}{x}^2=a^2+c^2-2ac\cos B\\ x^2={810}^2+{650}^2-2\left(810\right)\left(650\right)\cos 113°\\ x^2=1490039.88\\ x=\sqrt{1490039.88}\\ x=1220.67\\ x\approx 1221mi\end{array}$

Now, we will find angle $A$ ,

  $\begin{array}{l}\frac{a}{\sin A}=\frac{x}{\mathrm{sinB}}\\ \sin A=\frac{a\sin B}{x}\\ \sin A=\frac{810\sin 113°}{1221}\\ \sin A=0.61\\ A={\sin }^{-1}0.61\\ A=37.64°\end{array}$

Hence,

  $\begin{array}{l}\theta =48°+37.64°\\ \theta =\boxed{85.6°}\end{array}$

Hence, the distance from city $A$ to city $C$ and the bearing from city $A$ to city $C$ is $\boxed{x\approx 1221miles}$ , $\boxed{\theta =85.6°}$.