Chapter 12.8, Problem 5P

To determine

Find the distance he is now from Vista, and the angle he has to turn to fly directly there.

Answer to Problem 5P

The distance he is now from Vista is about 232.1 km, and the angle he has to turn to fly directly there is about $76.3°$ .

Explanation of Solution

Given:

J is flying a plane from Midville to Vista , a distance of 500 km. Because of a thunderstorm , he had to fly $17.5°$ off course for 300 km.

Formula Used:

Law of Cosine :

  $a^2=b^2+c^2-2bc\cos A$

Law of Sine:

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

  

Calculation:

Represent the situation :

  

We need to find measure of angle A and b.

Use Law of Cosine:

  $\begin{array}{l}{b}^2=a^2+c^2-2ac\cos B\\ \Rightarrow b^2={300}^2+{500}^2-2(300)(500)\cos 17.5°\\ \Rightarrow b^2=340000-300000\cos 17.5°\\ \Rightarrow b\approx 232.1\end{array}$

Use Law of Sine:

  $\begin{array}{l}\frac{\sin A}{500}=\frac{\sin 17.5°}{232.1}\\ \Rightarrow \sin A=500\cdot \frac{\sin 17.5°}{232.1}\mathrm{.................}[\text{multiply each side by }500]\\ \Rightarrow A\approx 40.4°\\ \Rightarrow A=180°-48°-55.7°=76.3°\end{array}$

So, the distance he is now from Vista is about 232.1 km, and the angle he has to turn to fly directly there is about $76.3°$ .