An airplane travels 6200 km against the wind in 8 hours and 7080 km with the wind in the same amount of time. What is the rate of the plane in still air and what is the rate of the wind?
6200/(p-w) = 8
7080/(p+w) = 8
(p,w) = (830,55)
To solve this problem, we can use the concept of relative velocities. Let's assume the rate of the plane in still air (the speed without any wind) is 'p' km/h, and the rate of the wind is 'w' km/h.
When the plane is flying against the wind, its effective speed (relative to the ground) decreases. So, the actual speed of the plane against the wind can be calculated by subtracting the speed of the wind from the rate of the plane in still air: (p - w) km/h.
Similarly, when the plane is flying with the wind, its effective speed increases. So, the actual speed of the plane with the wind can be calculated by adding the speed of the wind to the rate of the plane in still air: (p + w) km/h.
According to the given information:
- The plane traveled 6200 km against the wind in 8 hours, so we have the equation: 6200 = 8(p - w).
- The plane traveled 7080 km with the wind in the same amount of time, so we have the equation: 7080 = 8(p + w).
We have a system of two equations with two variables. Let's solve it:
From the first equation, divide both sides by 8: (p - w) = 6200/8 = 775.
From the second equation, divide both sides by 8: (p + w) = 7080/8 = 885.
Now, we can solve this system of equations to find the values of 'p' and 'w'.
Adding the two equations together, we get: (p - w) + (p + w) = 775 + 885.
This simplifies to: 2p = 1660.
Divide both sides by 2: p = 830.
Substituting the value of 'p' back into one of the equations, we can find 'w':
(p + w) = 885.
(830 + w) = 885.
Subtract 830 from both sides: w = 885 - 830.
w = 55.
Therefore, the rate of the plane in still air is 830 km/h, and the rate of the wind is 55 km/h.