SOLUTION: Sandy made a trip to a city 200 miles away and then returned home. Her average speed on the return trip was 10 mph less than her average speed going. If her total travel time was 9

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Question 1034960: Sandy made a trip to a city 200 miles away and then returned home. Her average speed on the return trip was 10 mph less than her average speed going. If her total travel time was 9 hours, what was her average rate in each direction?
Found 2 solutions by Alan3354, addingup:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Sandy made a trip to a city 200 miles away and then returned home. Her average speed on the return trip was 10 mph less than her average speed going. If her total travel time was 9 hours, what was her average rate in each direction?
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r = speed going
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d = r*t
t = d/r
200/r + 200/(r-10) = 9
200(r-10) + 200r = 9r(r-10)
400r - 2000 = 9r^2 - 90r
9r^2 - 490r + 1910 = 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-490x%2B1910+=+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=%28-490%29%5E2-4%2A9%2A1910=171340.

Discriminant d=171340 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--490%2B-sqrt%28+171340+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-490%29%2Bsqrt%28+171340+%29%29%2F2%5C9+=+50.218464534692
x%5B2%5D+=+%28-%28-490%29-sqrt%28+171340+%29%29%2F2%5C9+=+4.22597990975241

Quadratic expression 9x%5E2%2B-490x%2B1910 can be factored:
9x%5E2%2B-490x%2B1910+=+%28x-50.218464534692%29%2A%28x-4.22597990975241%29
Again, the answer is: 50.218464534692, 4.22597990975241. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+9%2Ax%5E2%2B-490%2Ax%2B1910+%29

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r = 50.22 going, 40.22 returning
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Ignore the 4... solution, subtracting 10 would give a negative speed.

Answer by addingup(3677) About Me  (Show Source):
You can put this solution on YOUR website!
200 out
200 back
9 hrs total
d= st
d/s = t
200%2Fx%2B200%2F%28x-10%29+=+9
400%28x-5%29%2Fx%28x-10%29+=+9
400%28x-5%29+=+9x%28x-10%29
400x-2000+=+9x%28x-10%29
400x-2000+=+9x%5E2-90
9x%5E2%2B490x-2000+=+0%0D%0A%7B%7B%7B-%28%28x-50%29%289x-40%29%29=+0
Multiply times -1
%28x-50%29%289x-40%29=+0
x-50+=+0 or 9x-40+=+0
x+=+50 or 9x+=+40; x+=+40%2F9
So, on the way out her speed was 50mph and on the return 50-10=40
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Check:
50+40 = 90; 90/2 = 45; 45*9 = 405 it should be 400, the 5 is the result of rounding numbers. So the answer is correct.