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: .$\displaystyle \frac{3000\text{ km}}{1.433\text{ hrs}} \;\approx\;2094\text{ km/hr}$