Hello, Frostking!

An airplane is flying at 500km/hr in a wind blowing 60 km/hr toward the SE.

In what direction should the plane head to end up going due east?

What is the plane's speed relative to the ground? Code:

B
*
* | *
500 * | * 60
* |h *
* | *
* θ | 45° *
A * * * * * * * * * * C
: - - - - - - - - b - - - - - - - - :

The plane flies from $\displaystyle A$ to $\displaystyle B$ at 500 km/hr.

The wind is blowing from $\displaystyle B$ to $\displaystyle C$ at 60 m/hr.

The result flight is $\displaystyle AC$, directly east.

In the right triangle at the right: .$\displaystyle \sin45^o \:=\:\frac{h}{60}\quad\Rightarrow\quad h \:=\:60\sin45^o$ [1]

In the right triangle at the left: .$\displaystyle \sin\theta \:=\:\frac{h}{500}$ [2]

Substitute [1] into [2]: .$\displaystyle \sin\theta \:=\:\frac{60\sin45^o}{500} \:=\:0.984852814$

Hence: .$\displaystyle \theta \;=\;4.867561141^o$

The direction should be about: .$\displaystyle 4.868^o$ north of east.

We have: .$\displaystyle \angle B \:=\:180^o - 45^o - 4.868^o \:=\:130.132^o$

Law of Sines: .$\displaystyle \frac{b}{\sin130.132^o} \:=\:\frac{500}{\sin45^o} \quad\Rightarrow\quad b \:=\:540.6266461$

Its ground speed is about: .$\displaystyle 540.63$ km/hr.