Question

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.

Answer 1

`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`