Question

An airplane flies at 400 miles per hour due east in a wind blowing at 50 miles per hour
from southeast. What is the flying speed and direction of the plane relative to the ground?

Answer 1

Question

The speed is $v = \sqrt{50^2 + 400^2 - 2 \cdot 50 \cdot 400 \cdot \cos 45{}^\circ} \approx 366$ miles per hour.

Direction:

$$\frac{366}{\sin 45{}^\circ} = \frac{50}{\sin \alpha} \Rightarrow \sin \alpha = \frac{50 \sin 45{}^\circ}{366} \Rightarrow \alpha = \arcsin \left( \frac{50 \sin 45{}^\circ}{366} \right) \approx 5.54{}^\circ \text{ north of east}.$$

Answer: speed is nearly 366 miles per hour and direction is 5.54 degree north of east.