SOLUTION: An airplane flies 250 km against the wind and 680 km with the wind in a total time of 4 hour. The speed of the airplane is 300 km/h. What is the speed of the wind?

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Question 1031838: An airplane flies 250 km against the wind and 680 km with the wind in a total time of 4 hour. The speed of the airplane is 300 km/h. What is the speed of the wind?
Found 2 solutions by LinnW, Alan3354:
Answer by LinnW(1048) About Me  (Show Source):
You can put this solution on YOUR website!
Set w = the wind speed.
The time t1 for the 250 km leg is the distance 250 / the speed (300 - w)
The time t2 for the 680 km leg is the distance 680 / the speed (300 + w)
t1 + t2 = 4 hours
250%2F%28300-w%29%2B680%2F%28300%2Bw%29=4
w = 205.86 is the wind speed.
Solving completely by hand is fairly tedious.
I used www.wolframalpha.com with the above equation.
If your instructor needs a more detailed solution,
let me know.

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
An airplane flies 250 km against the wind and 680 km with the wind in a total time of 4 hour. The speed of the airplane is 300 km/h. What is the speed of the wind?
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a = plane's airspeed = 300
w = windspeed
----
250/(300-w) + 680/(300+w) = 4
250*(300+w) + 680*(300-w) = 4*(300+w)*(300-w) = 360000 - 4w^2
75000 + 204000 - 430w = 360000 - 4w^2
4w^2 - 430w - 81000 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 4x%5E2%2B-430x%2B-81000+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-430%29%5E2-4%2A4%2A-81000=1480900.

Discriminant d=1480900 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--430%2B-sqrt%28+1480900+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-430%29%2Bsqrt%28+1480900+%29%29%2F2%5C4+=+205.865293445465
x%5B2%5D+=+%28-%28-430%29-sqrt%28+1480900+%29%29%2F2%5C4+=+-98.3652934454652

Quadratic expression 4x%5E2%2B-430x%2B-81000 can be factored:
4x%5E2%2B-430x%2B-81000+=+%28x-205.865293445465%29%2A%28x--98.3652934454652%29
Again, the answer is: 205.865293445465, -98.3652934454652. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+4%2Ax%5E2%2B-430%2Ax%2B-81000+%29

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w =~ 206 km/hr
======================
No fun flying upwind like that.
Downwind is great.