SOLUTION: Miami and Pittsburg are 1000 miles apart. A plane flew into a 50mph headwind from miami to pittsburg. on the return flight the 50mph wind became a tailwind. The lane was in the air

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Question 750065: Miami and Pittsburg are 1000 miles apart. A plane flew into a 50mph headwind from miami to pittsburg. on the return flight the 50mph wind became a tailwind. The lane was in the air a total of 4.5 hrs. for the roundtrip. what would have been the planes average speed without the wind?
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Miami and Pittsburg are 1000 miles apart. A plane flew into a 50mph headwind from miami to pittsburg. on the return flight the 50mph wind became a tailwind. The lane was in the air a total of 4.5 hrs. for the roundtrip. what would have been the planes average speed without the wind?
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p = plane's airspeed
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1000/(p - 50) + 1000/(p + 50) = 4.5
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1000*(2p) = 4.5*(p^2 - 2500)
4000p = 9p^2 - 22500
9p^2 - 4000p - 22500 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 9x%5E2%2B-4000x%2B-22500+=+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=%28-4000%29%5E2-4%2A9%2A-22500=16810000.

Discriminant d=16810000 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--4000%2B-sqrt%28+16810000+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-4000%29%2Bsqrt%28+16810000+%29%29%2F2%5C9+=+450
x%5B2%5D+=+%28-%28-4000%29-sqrt%28+16810000+%29%29%2F2%5C9+=+-5.55555555555556

Quadratic expression 9x%5E2%2B-4000x%2B-22500 can be factored:
9x%5E2%2B-4000x%2B-22500+=+%28x-450%29%2A%28x--5.55555555555556%29
Again, the answer is: 450, -5.55555555555556. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+9%2Ax%5E2%2B-4000%2Ax%2B-22500+%29

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Ignore the negative solution,
p = 450 mi/hr
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with no wind, the avg = 450 mi/hr