SOLUTION: Thank you in advance for your help. I have attacked this problem many different ways and still come up short. The problem is: The members of a bicycle club rode a 20-mile round

Algebra ->  Rate-of-work-word-problems -> SOLUTION: Thank you in advance for your help. I have attacked this problem many different ways and still come up short. The problem is: The members of a bicycle club rode a 20-mile round       Log On


   



Question 305501: Thank you in advance for your help. I have attacked this problem many different ways and still come up short. The problem is:
The members of a bicycle club rode a 20-mile round trip route. On the way back, they had a tailwind and averaged 3 mph faster than on the first 10 miles of the trip. If the entire trip took them 1 1/2 hours, what was their average speed on the first half of the ride?

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
The members of a bicycle club rode a 20-mile round trip route. On the way back, they had a tailwind and averaged 3 mph faster than on the first 10 miles of the trip. If the entire trip took them 1 1/2 hours, what was their average speed on the first half of the ride?
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d = distance one way = 10 miles
r = rate going without tailwind
d/r + d/(r+3) = 1.5
10/r + 10/(r+3) = 1.5
10(r+3) + 10r = 1.5r(r+3)
20r+%2B+30+=+1.5r%5E2+%2B+4.5r
3r%5E3+-+31r+-+60+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 3x%5E2%2B-31x%2B-60+=+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=%28-31%29%5E2-4%2A3%2A-60=1681.

Discriminant d=1681 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--31%2B-sqrt%28+1681+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-31%29%2Bsqrt%28+1681+%29%29%2F2%5C3+=+12
x%5B2%5D+=+%28-%28-31%29-sqrt%28+1681+%29%29%2F2%5C3+=+-1.66666666666667

Quadratic expression 3x%5E2%2B-31x%2B-60 can be factored:
3x%5E2%2B-31x%2B-60+=+%28x-12%29%2A%28x--1.66666666666667%29
Again, the answer is: 12, -1.66666666666667. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B-31%2Ax%2B-60+%29

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Ignore the negative solution
r = 12 mph (1st half)