Question

The engines of a plane are pushing it due north at a rate of 300 mph, and the wind is pushing the plane 20° west of north at a rate of 40 mph. In what direction is the plane going?Round to the nearest tenth.

Answer 1

3° west of north

Explanation:

To better understand the question we can do the following sketch:

The first thing is to calculate the magnitude of the resulting vector, by means of the law of cosines, as follows:

`\begin{gathered} V_r=√((V_1)^2+(V_2)^2-2\cdot V_1\cdot V_2\cdot\cos(\theta)) \\ \\ \text{ We replacing:} \\ \\ V_r=√(300^2+40^2-2\cdot300\cdot40\cdot\:\cos\left(20°\right)) \\ \\ V_r=262.77 \end{gathered}`

Now, we can calculate the angle, using the law of sines, like this:

`\begin{gathered} (\sin\alpha)/(40)=(\sin20\degree)/(262.8) \\ \\ \sin\alpha=(\sin20\degree)/(262.8)\cdot40 \\ \\ \alpha=\sin^(-1)(0.0520) \\ \\ \alpha=2.98\degree=3.0\degree \end{gathered}`

The direction is 3° west of north