SOLUTION: I have been stuck on this one for a while now. Trying to define variables, write and equation and solve. Please help me!! The distance going up and down a hill is 15 miles. I

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Question 524424: I have been stuck on this one for a while now. Trying to define variables, write and equation and solve. Please help me!!
The distance going up and down a hill is 15 miles. It takes a total of 2 hours to complete the trip. Going down the hill is 20 mph faster than going up. How fast is the trip up the hill in mph?
Found 2 solutions by nerdybill, Alan3354:
Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
The distance going up and down a hill is 15 miles. It takes a total of 2 hours to complete the trip. Going down the hill is 20 mph faster than going up. How fast is the trip up the hill in mph?
Let y = time (hours) going uphill
then
2-y = time (hours) going downhill
.
Let x = speed (mph) going uphill
then
x+20 = speed (mph) going downhill
.
applying distance formula of d=rt
xy = 15 (equation 1)
(x+20)(2-y) = 15 (equation 2)
.
solving equation 1 for y:
y = 15/x
.
plug into equation 2 and solve for x:
(x+20)(2-y) = 15
(x+20)(2-15/x) = 15
multiplying both sides by x:
(x+20)(2x-15) = 15x
2x^2-15x+40x-300 = 15x
2x^2+25x-300 = 15x
2x^2+10x-300 = 0
x^2+5x-150 = 0
(x+15)(x-10) = 0
x = {-15, 10}
throw out negative solution leaving
x = 10 mph (speed going up the hill)

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
The distance going up and down a hill is 15 miles. It takes a total of 2 hours to complete the trip. Going down the hill is 20 mph faster than going up. How fast is the trip up the hill in mph?
----------
I think you mean it's 7.5 miles each way.
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r = speed going up
7.5/r + 7.5/(r+20) = 2
15/r + 15/(r+20) = 4 No fractions
Multiply thru by r(r+20)
15(r+20) + 15r = 4r*(r+20) = 4r^2 + 80r
30r + 300 = 4r^2 + 80r
4r^2 + 50r - 300 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 4x%5E2%2B50x%2B-300+=+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=%2850%29%5E2-4%2A4%2A-300=7300.

Discriminant d=7300 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-50%2B-sqrt%28+7300+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%2850%29%2Bsqrt%28+7300+%29%29%2F2%5C4+=+4.43000468164691
x%5B2%5D+=+%28-%2850%29-sqrt%28+7300+%29%29%2F2%5C4+=+-16.9300046816469

Quadratic expression 4x%5E2%2B50x%2B-300 can be factored:
4x%5E2%2B50x%2B-300+=+%28x-4.43000468164691%29%2A%28x--16.9300046816469%29
Again, the answer is: 4.43000468164691, -16.9300046816469. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+4%2Ax%5E2%2B50%2Ax%2B-300+%29

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Ignore the negative result
r =~ 4.43 mi/hr going up
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If it's 15 miles each way, the other tutor's answer is correct.
You should have made it clear.