I'm really in big trouble with this homework question. I can't seem to figure out how to tackle this problem.
Can somebody help out?
Thanks!
A plane is heading due south. If its airspeed is 530 km/hr and there is a wind blowing 90 km/hr to the northwest, what is the ground speed of the plane?
ground speed = ______
Best Answer
You have a triangle with two sides given, and the angle between them can be quickly figured out. Therefore, the Law of Cosines can be used to solve for the third side. The two sides of the triangle we know of are the plane's airspeed and the wind's speed. The angle between them is simply $45°$ because that is the angle between any vector in the northwest direction and any vector in the northward direction.
The Law of Cosines states that
$$c^2 = a^2 + b^2 - 2ab \cos C$$
We know $a$, $b$, and $C$, so we can plug into this formula to get
$$ c^2 = 530^2 + 90^2 - 2(530)(90)\cos 45°$$
Solving for $c$ yields the answer $c = \sqrt{289000 - 47700 \sqrt 2} $ $\frac{km}{hr}$, where $c ≈ 470.6824971$$\frac{km}{hr}$, which makes sense given the information in the problem.