An aircraft is flying at a speed of 2000km/h, wishes to reach a city 3000 km due East. Encounters wind blowing from N70W with speed of 100 km/h. Find the velocity of the plane relative to the ground. How long will it take to reach the city?

2. Hello, peon123!

A plane, flying at 2000 km/h, wishes to reach a city 3000 km due East,
encounters wind blowing from N70W with speed 100 km/h.
(a) Find the velocity of the plane relative to the ground.
(b) How long will it take to reach the city?
Code:
               D      N
*     :
*70°:
* :
oC
2000t    *   : *  100t
*       :   *
*              20°*
o - - - - - - - - - - - o
A           3000          B

The plane intends to fly from $\displaystyle A$ to $\displaystyle B$: .$\displaystyle AB = 3000$ km.

The wind is blowing from D to B: .$\displaystyle \angle DCN = 70^o$
. . Hence: .$\displaystyle \angle CBA = 20^p$

Because of the wind, the plane flies from A to C at 2000 km/hr.
. . In $\displaystyle t$ hours, it flies 2000t km: .$\displaystyle AC = 2000t$

In the same $\displaystyle t$ hours, the wind blows from C to B: .$\displaystyle CB = 100t$ km.

In $\displaystyle \Delta ABC$, Law of Cosines: .$\displaystyle (2000t)^2 \:=\:3000^2+(100t)^2 - 2(3000)(100t)\cos20^o$

This simplifies to: .$\displaystyle 399t^2 + 56.382t - 900 \:=\:0$

Quadratic Formula: .$\displaystyle t \;=\;\frac{-56.382 \pm\sqrt{56.382^2 + 4(399)(900)}}{2(399)}$

. . which has the positive root: .$\displaystyle t \:\approx\:1.433$

(b) The flight took 1.433 hours.

(a) The plane's speed was: .$\displaystyle \frac{3000\text{ km}}{1.433\text{ hrs}} \;\approx\;2094\text{ km/hr}$