Use Law of Cosines: Two airplanes leave an airport at the same time. The first flies 150 km/h in a direction of #320^@#. The second flies 200 km/h in a direction of #200^@#. After 3 hr, how far apart are the planes?

Answer is #~~# 912 km.

Thanks in advance.

1 Answer
Harish Chandra Rajpoot
Jul 26, 2018

#912.414\ \text{km#

Explanation:

The angle #\theta# between the directions of motion of planes

#\theta=320-200#

#=120^\circ#

The distance #a# traveled by first plane flying at a speed of #150\ \text{km/hr}# in #3\ \text{hrs#

#a=150\cdot 3#

#=450\ km#

Similarly, the distance #b# traveled by second plane flying at a speed of #200\ \text{km/hr}# in #3\ \text{hrs#

#b=200\cdot 3#

#=600\ km#

Now, the distance between the planes after #3# hrs will be equal to the side of a triangle opposite to the angle #\theta=120^\circ# included by the sides #a=450# & #b=600 #.

Now, using Cosine rule in the concerned triangle, the distance between the plane after #3# hours is given as

#\sqrt{a^2+b^2-2a\cos\theta}#

#=\sqrt{450^2+600^2-2\cdot 450\cdot 600\cos120^\circ}#

#=\sqrt{562500-540000(-1/2)}#

#=150\sqrt37#

#=912.414\ \text{km#

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