An airplane encountered a head wind during a flight between Rontown and Sawburgh which took 4 hours and 24 minutes. The return flight took 4 hours. If the distance from Rontown to Sawburgh is 1700 miles, find the airspeed of the plane (the speed of the plane in still air) and the speed of the wind, assuming both remain constant.
assuming the plane flies at speed p, and the wind at speed w,
4.4(s-w) = 4(s+w)
4.4s - 4.4w = 4s + 4w
.4s = 8.4w
s = 21w
4*22w = 1700
w = 19.318
s = 405.68
check:
4.4*386.36 = 1700
4*425 = 1700
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To solve this problem, we can use the concept of relative speed. Let's assume the airspeed of the plane (speed in still air) is 'x' miles per hour, and the speed of the wind is 'y' miles per hour.
During the flight from Rontown to Sawburgh, the plane encountered a headwind, which means its effective speed reduced. The time taken for this part of the journey is 4 hours and 24 minutes, which is equivalent to 4.4 hours (since 24 minutes is 0.4 of an hour).
To find the airspeed of the plane, we need to consider the relative speed between the plane and the wind. The formula for relative speed is:
Relative Speed = Speed of the Plane - Speed of the Wind
We know that the distance from Rontown to Sawburgh is 1700 miles, and the time taken is 4.4 hours. So, we can set up the following equation:
1700 / 4.4 = x - y
Now, let's consider the return flight from Sawburgh to Rontown, which took 4 hours. During this flight, the plane would have a tailwind, which means its effective speed would increase. Again, using the formula for relative speed, we have:
1700 / 4 = x + y
We now have two equations:
1700 / 4.4 = x - y (Equation 1)
1700 / 4 = x + y (Equation 2)
To solve this system of equations, we can eliminate the variable 'y' by adding Equation 1 and Equation 2:
(1700 / 4.4) + (1700 / 4) = (x - y) + (x + y)
Simplifying:
(1700 * 4 + 4.4 * 1700) / 4.4 = 2x
Multiplying through by 4.4:
(6800 + 7480) / 4.4 = 2x
14280 / 4.4 = 2x
3245.45 = 2x
Dividing both sides by 2:
x = 3245.45 / 2
x = 1622.73
Therefore, the airspeed of the plane (speed in still air) is approximately 1622.73 miles per hour.
To find the speed of the wind, we can substitute the value of x into Equation 1:
1700 / 4.4 = 1622.73 - y
Simplifying:
1700 = 4.4 * 1622.73 - 4.4y
7480 = 7141.39 - 4.4y
Subtracting 7141.39 from both sides:
7480 - 7141.39 = -4.4y
338.61 = -4.4y
Dividing by -4.4:
y = -338.61 / -4.4
y ≈ 76.91
Therefore, the speed of the wind is approximately 76.91 miles per hour.
To find the airspeed of the plane and the speed of the wind, we can use the formula:
Distance = Speed * Time
Let's denote the airspeed of the plane as P and the speed of the wind as W.
For the flight from Rontown to Sawburgh, the total time (including the effect of the headwind) is 4 hours and 24 minutes, which is equivalent to 4.4 hours.
Using the given information, we can set up the following equations:
1. For the flight from Rontown to Sawburgh:
Distance = (P - W) * 4.4
2. For the return flight from Sawburgh to Rontown:
Distance = (P + W) * 4
Since the distance is the same in both cases (1700 miles), we can equate the two equations:
(P - W) * 4.4 = (P + W) * 4
Expanding the equation:
4.4P - 4.4W = 4P + 4W
Collecting like terms:
4.4P - 4P = 4W + 4.4W
0.4P = 8.4W
Dividing both sides by 0.4:
P = 21W
Now, we have a relationship between the airspeed of the plane (P) and the speed of the wind (W).
Substituting P = 21W into either of the original equations, let's use the first equation:
1700 = (21W - W) * 4.4
Expanding the equation:
1700 = 20W * 4.4
Dividing both sides by 20 * 4.4:
W = 1700 / (20 * 4.4) ≈ 19.32 mph (rounded to two decimal places)
Now, substitute the value of W back into P = 21W to find the airspeed of the plane:
P = 21 * 19.32 ≈ 405.72 mph (rounded to two decimal places)
Therefore, the airspeed of the plane (speed in still air) is approximately 405.72 mph and the speed of the wind is approximately 19.32 mph.