SOLUTION: Trying to help my son with Algebra II, and I just can't remember! Plane flies roundtrip from Denver to San Francisco a total of 1600 miles. Trip time is 9 hours. Return trip f

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Question 342957: Trying to help my son with Algebra II, and I just can't remember!
Plane flies roundtrip from Denver to San Francisco a total of 1600 miles. Trip time is 9 hours. Return trip from San Francisco, the plane averages 40 mph faster than the outbound flight. What is the average speed to San Fran and back?
X1 = Avg speed out in mph
X2 = Avg speed back in mph
Y1 = time out in hours
Y2 = time back in hours
1600 Miles = X1*Y1 + X2*Y2
Y2= 9-Y1
X2 = X1 + 40 mph
X1 = 800/Y1
I set up to solve for mph, using the substitutions I've identified, and get lost...
Found 2 solutions by Alan3354, solver91311:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Plane flies roundtrip from Denver to San Francisco a total of 1600 miles. Trip time is 9 hours. Return trip from San Francisco, the plane averages 40 mph faster than the outbound flight. What is the average speed to San Fran and back?
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Is 9 hours the time for the round trip? 800 miles each way? If so:
The average overall is 1600/9 =~ 177.8 mph
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If you want the avg speed for each half of the trip:
d = rt
t = d/r
9 = 800/r + 800/(r+40)
9*r*(r+40) = 800(r+40) + 800r
9r^2 + 360r = 800r + 3200 + 800r = 1600r + 3200
9r^2 - 1240r - 3200 = 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-1240x%2B-3200+=+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=%28-1240%29%5E2-4%2A9%2A-3200=1652800.

Discriminant d=1652800 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--1240%2B-sqrt%28+1652800+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-1240%29%2Bsqrt%28+1652800+%29%29%2F2%5C9+=+140.311816355503
x%5B2%5D+=+%28-%28-1240%29-sqrt%28+1652800+%29%29%2F2%5C9+=+-2.53403857772532

Quadratic expression 9x%5E2%2B-1240x%2B-3200 can be factored:
9x%5E2%2B-1240x%2B-3200+=+%28x-140.311816355503%29%2A%28x--2.53403857772532%29
Again, the answer is: 140.311816355503, -2.53403857772532. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+9%2Ax%5E2%2B-1240%2Ax%2B-3200+%29

r = x (Ignore the negative solution)
r =~ 140 going
r + 40 =~ 180 returning

Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


You are making this too complex in your setup. Try it this way:

Let represent the rate of speed on the trip from Denver to SF. Let represent the amount of elapsed time for the Denver to SF trip. Then you can say that the return trip was at mph and took hours. Furthermore, since SF and Denver are, relatively speaking, stationary, if the total round trip distance is 1600 miles, it must be 800 miles each way.

Now we can describe the outbound trip (in terms of distance equals rate times time) as:



And the return trip is:



Solve each of the above for in terms of everything else:



and







Now you have two expressions both equal to . Set them equal to each other.



Cross-multiply and collect like terms in the LHS, setting everything in the left equal to zero:



Solve the factorable quadratic for for the average rate of speed on the trip to SF. Obviously, you will discard any negative root. Add 40 to get the average rate of speed on the return to Denver.

John

My calculator said it, I believe it, that settles it