A plane traveled 400 miles to Rome and back. The trip there was with the wind. It took 5 hours. The trip back was into the wind. The trip back took 10 hours.
- Find the speed of the plane in still air and the speed of the wind.
Assuming that each way is 400 miles, (even though in Rome they wouldn't know what miles are, and would be using km.)
let speed of plane in still air be x mph
let speed of wind be y mph
going with the wind:
5(x+y) = 400 ---> x + y = 80
going against the wind:
10(x-y) = 400 ---> x - y = 40
add them:
2x = 120
x = 60
then y = 20
the speed of the plane is 60 mph, and the wind has a speed of 20 mph.
(that's a pretty slow plane)
how did you get y=20
To find the speed of the plane in still air and the speed of the wind, we can set up a system of equations using the given information.
Let's assume the speed of the plane in still air is represented by "p" and the speed of the wind is represented by "w".
1. On the trip there:
The distance traveled is 400 miles and it took 5 hours. Since the plane was traveling with the wind, we can write the equation: Distance = Speed * Time
400 = (p + w) * 5 ----(Equation 1)
2. On the trip back:
The distance traveled is still 400 miles, but it took 10 hours. Since the plane was traveling against the wind, the effective speed will be reduced by the wind's speed. Thus, we can write: Distance = Speed * Time
400 = (p - w) * 10 ----(Equation 2)
Now, we have a system of two equations (Equation 1 and Equation 2) that we can solve simultaneously to find the values of p (speed of the plane in still air) and w (speed of the wind).
Let's simplify the equations and solve the system:
From Equation 1:
400 = 5p + 5w ----(Equation 1')
From Equation 2:
400 = 10p - 10w ----(Equation 2')
Now, we have a system of two linear equations:
5p + 5w = 400 ----(Equation 1')
10p - 10w = 400 ----(Equation 2')
We can solve this system using different methods such as substitution, elimination, or matrix operations. Let's solve it using the elimination method:
Multiply Equation 1' by 2 to eliminate the "w" term:
10p + 10w = 800
Now, add Equation 2' to this equation:
10p - 10w + (10p + 10w) = 400 + 800
20p = 1200
Divide both sides by 20:
p = 60
Substitute the value of p = 60 into Equation 1' or Equation 2' to find the value of w:
5(60) + 5w = 400
300 + 5w = 400
5w = 400 - 300
5w = 100
w = 20
Therefore, the speed of the plane in still air (p) is 60 miles per hour, and the speed of the wind (w) is 20 miles per hour.