Flying against the wind, an airplane travels 28 km in 5 hours. Traveling with the win, the same plane travels 3760km in 4 hours. What is the rate of the plane and still air and what is the rate of the wind?
To find the rate of the plane in still air (its speed without the influence of the wind) and the rate of the wind itself, we can set up a system of equations.
Let's denote the rate of the plane in still air as "p" and the rate of the wind as "w".
When flying against the wind, the effective speed of the plane is reduced. In this case, the plane covers 28 km in 5 hours. Therefore, the equation for this scenario is:
(p - w) * 5 = 28 -- Equation 1
When flying with the wind, the plane's effective speed increases. In this case, the plane covers 3760 km in 4 hours. Therefore, the equation for this scenario is:
(p + w) * 4 = 3760 -- Equation 2
Now, we have a system of two equations:
(p - w) * 5 = 28 -- Equation 1
(p + w) * 4 = 3760 -- Equation 2
We can solve this system of equations to find the values of "p" and "w".
First, let's solve Equation 1 for "p - w":
(p - w) = 28/5
Simplifying, we get:
p - w = 5.6 -- Equation 3
Now, let's solve Equation 2 for "p + w":
(p + w) = 3760/4
Simplifying, we get:
p + w = 940 -- Equation 4
Now we have a system of equations:
Equation 3: p - w = 5.6
Equation 4: p + w = 940
To solve this system, we can add Equations 3 and 4 together:
(p - w) + (p + w) = 5.6 + 940
This simplifies to:
2p = 945.6
Dividing both sides by 2, we get:
p = 945.6/2
p = 472.8 km/h
Now, we can substitute the value of p into Equation 4 to find the value of w:
472.8 + w = 940
Subtracting 472.8 from both sides, we get:
w = 940 - 472.8
w = 467.2 km/h
So, the rate of the plane in still air is 472.8 km/h, and the rate of the wind is 467.2 km/h.