An airplane flying with the wind can cover a certain distance in a = 3 hours. The return trip against the wind takes b = 4.5 hours. How fast is the plane and what is the speed of the wind, if the one-way distance is c = 720 miles?
speed of plane in no wind --- x mph
speed of wind ---- y mph
time with the wind = 720/(x+y) = 3
3x + 3y = 720
x + y = 240
time against the wind = 720/(x-y) = 4
4x - 4y = 720
x-y = 180
add the two simplified equations:
2x = 420
x = 210
back in x+y=240
210 + y = 240
y = 30
the plane has a speed of 210 mph, and the wind was 30 mph
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To find the speed of the plane and the speed of the wind, we can set up a system of equations.
Let's denote the speed of the plane as "p" and the speed of the wind as "w".
For the journey with the wind, the effective speed of the plane would be (p + w). Therefore, the distance covered in 3 hours can be calculated using the formula:
distance = speed × time
720 = (p + w) × 3
For the return journey against the wind, the effective speed of the plane would be (p - w). Therefore, the distance covered in 4.5 hours can be calculated using the formula:
distance = speed × time
720 = (p - w) × 4.5
Now, we can set up a system of equations:
1) 720 = (p + w) × 3
2) 720 = (p - w) × 4.5
Let's solve this system of equations to find the values of p and w:
Step 1: Simplify Equation 1 and Equation 2
1) 240 = p + w
2) 160 = p - w
Step 2: Add Equation 1 and Equation 2
240 + 160 = (p + w) + (p - w)
400 = 2p
Step 3: Divide both sides by 2 to solve for p
400/2 = p
p = 200
Step 4: Substitute the value of p back into Equation 1 or Equation 2 to solve for w
240 = 200 + w
w = 240 - 200
w = 40
Therefore, the speed of the plane is 200 miles per hour, and the speed of the wind is 40 miles per hour.
To determine the speed of the plane and the speed of the wind, we need to set up a system of equations using the given information.
Let's assume that the speed of the plane is represented by "p" (in miles per hour) and the speed of the wind is represented by "w" (in miles per hour).
When the plane is flying with the wind, the effective speed of the plane will be the sum of its actual speed and the speed of the wind. Therefore, the equation for the distance covered with the wind can be written as:
c = (p + w) * a.....(1)
Similarly, when the plane is flying against the wind, the effective speed of the plane will be the difference between its actual speed and the speed of the wind. Hence, the equation for the return trip against the wind can be written as:
c = (p - w) * b.....(2)
We are given that a = 3 hours, b = 4.5 hours, and c = 720 miles.
Substituting the given values into equations (1) and (2), we get:
720 = (p + w) * 3.....(3)
720 = (p - w) * 4.5.....(4)
Now we have a system of two equations with two variables. We can solve this system to find the values of p and w.
Divide equation (3) by 3:
240 = p + w.....(5)
Divide equation (4) by 4.5:
160 = p - w.....(6)
Now we have a system of two equations:
240 = p + w.....(5)
160 = p - w.....(6)
To solve this system, we can add equations (5) and (6):
240 + 160 = (p + p) + (w - w)
400 = 2p
Divide both sides of the equation by 2:
200 = p
Therefore, the speed of the plane is 200 miles per hour.
To find the speed of the wind, we will substitute the value of p back into equation (5):
240 = 200 + w
Subtract 200 from both sides:
40 = w
Therefore, the speed of the wind is 40 miles per hour.
So, the plane's speed is 200 mph and the speed of the wind is 40 mph.