Answer:
the speed of the plane in still air is 540 miles per hour and the speed of the wind is 280 miles per hour.
Explanation:
Step 1: Define the variables
Let's call the speed of the plane in still air "V". Let's call the speed of the wind "W".
Step 2: Write an equation for the distance traveled with the wind
The distance traveled by the plane with the wind in 6.5 hours can be represented by the equation:
D1 = (V + W) * t
where D1 is the distance traveled, t is the time (6.5 hours), and V and W are the speeds of the plane and the wind, respectively.
Substitute the known values into the equation:
3900 = (V + W) * 6.5
Step 3: Write an equation for the distance traveled against the wind
The distance traveled by the plane against the wind in 6.5 hours can be represented by the equation:
D2 = (V - W) * t
where D2 is the distance traveled, t is the time (6.5 hours), and V and W are the speeds of the plane and the wind, respectively.
Substitute the known values into the equation:
3120 = (V - W) * 6.5
Step 4: Solve for the speed of the wind
We can now use the two equations from Steps 2 and 3 to solve for the speed of the wind.
First, let's rearrange the equation from Step 2 to isolate "W":
W = (3900 / 6.5) - V
Next, substitute this expression for "W" into the equation from Step 3:
3120 = (V - ((3900 / 6.5) - V)) * 6.5
Expand the right side of the equation:
3120 = (V - 3900 / 6.5 + V) * 6.5
Simplify the equation:
3120 = 2V * 6.5 - 3900
Divide both sides of the equation by 2:
1560 = V * 6.5 - 1950
Multiply both sides of the equation by 2:
3120 = V * 13 - 3900
Add 3900 to both sides of the equation:
7020 = V * 13
Divide both sides of the equation by 13:
V = 540
Step 5: Solve for the speed of the wind
Now that we know the speed of the plane in still air, we can use the equation from Step 2 to find the speed of the wind:
W = (3900 / 6.5) - V
Substitute the value of "V" that we found in Step 4 into the equation:
W = (3900 / 6.5) - 540
Simplify the equation:
W = 280
Step 6: Conclusion
Therefore, the speed of the plane in still air is 540 miles per hour and the speed of the wind is 280 miles per hour.