The rate of the wind is 38 km/h and the rate of the airplane in still air is 152 km/h.
To find the rate of the wind and the rate of the airplane in still air, we need to use the concept of relative velocity.
Let's assume the rate of the airplane in still air is 'a' km/h and the rate of the wind is 'w' km/h.
For the first leg of the trip with a tailwind, the airplane's rate relative to the ground is (a + w) km/h. Given that the distance is 570 km and the time is 3 hours, we can write the equation: (a + w) * 3 = 570.
Solving this equation, we get a + w = 190.
For the return trip against the wind, the airplane's rate relative to the ground is (a - w) km/h. Given that the distance is 570 km and the time is 5 hours, we can write the equation: (a - w) * 5 = 570.
Solving this equation, we get a - w = 114.
Now, we have a system of two equations with two variables:
a + w = 190
a - w = 114
By adding the two equations together, we get 2a = 304, which gives us a = 152.
Substituting the value of a into one of the equations, we can solve for w:
152 + w = 190
w = 190 - 152
w = 38.
Therefore, the rate of the wind is 38 km/h and the rate of the airplane in still air is 152 km/h.