On a round trip of 400km,a pilot wants to determine the airspeed of hiis light plane. He flies the entire distance at the same airspeed for the first 200km a steady headwind slows him down by 30km/hour. the entire trips takes 1 1/2hours. Find the planes airspeed?

V will represent plane's airspeed.


We can divide pilot's trip in 2 parts:

1. first 200km, when he is flying with speed v. The time it takes him to fly this part of trip is t=200/v
2. last 200km, when he is flying with speed v-30. The time it takes him to fly this part is 200/(v-30)

The time it takes him for second part of trip is also equal to 1.5-t (because entire trip takes 1.5h), so now we can write equation:

1.5-t=200/(v-30)

We also know that t=200/v so we substitute that into equation:

1.5-200/v=200/(v-30)

To solve this equation we multiply it with v*(v-30)=v^2-30v first. We get:

1.5(v^2-30v)-200(v-30)=200v

Now we get rid of brackets:

1.5v^2-45v-200v+600-200v=0

or

1.5v^2-445v+600=0

This is quadratic equation, so we get 2 solutions (sqrt= square root):

1. v=(445+sqrt(445^2-4*1.5*600))/(2*1.5)=295.3 km/h
2. v=(445-sqrt(445^2-4*1.5*600))/(2*1.5)=1.4 km/h

We know that only v=295 km/h is solution because if 1.4 km/h was solution that would mean that second part of trip plane was flying with speed 1.4-30= -28.6 km/h, so that would mean he was flying backwards (which we know he didn't because it's round trip). Speed 1.4 km/h is also not physical solution, because plane wouldn't be able to take off with such small speed :)