Traveling with the wind, an airplane travels 210 miles im 4 hours. Against the wind, it takes six hours. Find the rate of the plane in still air and the velocity of the wind.

210 = 4 (v+w)

210 = 6 (v-w)

105 = 2 v + 2 w
105 = 3 v - 3 w

315 = 6 v + 6 w
210 = 6 v - 6 w
--------------------add
525 = 12 v etc

To solve this problem, we'll use the concept of relative speed. Let's denote the rate of the airplane in still air as "P" and the velocity of the wind as "W".

When the airplane is traveling with the wind, its effective speed is the sum of its rate in still air and the velocity of the wind. So, the speed of the airplane with the wind is P + W.

Similarly, when the airplane is flying against the wind, its effective speed is the difference between its rate in still air and the velocity of the wind. Therefore, the speed of the airplane against the wind is P - W.

Given that the airplane travels 210 miles in 4 hours with the wind, we can form the equation:

Distance = Speed × Time

210 = (P + W) × 4

Simplifying this equation, we get:

210 = 4P + 4W

Now, let's consider the airplane's speed against the wind. The airplane takes six hours to cover the same distance, so we can set up another equation:

210 = (P - W) × 6

Simplifying this equation, we have:

210 = 6P - 6W

Now, we have a system of equations:

4P + 4W = 210
6P - 6W = 210

To solve this system, we can use the method of substitution or elimination. Let's use the elimination method:

Multiplying the first equation by 3 and the second equation by 2 to eliminate the W term, we get:

12P + 12W = 630
12P - 12W = 420

Adding the equations together, we have:

24P = 1050

Dividing both sides by 24, we find:

P = 43.75

Now that we have the value of P (the rate of the airplane in still air), we can substitute this value into one of the original equations to solve for W.

Using the first equation:

4(43.75) + 4W = 210

175 + 4W = 210

4W = 35

Dividing both sides by 4, we find:

W = 8.75

Therefore, the rate of the airplane in still air (P) is 43.75 miles per hour, and the velocity of the wind (W) is 8.75 miles per hour.