SOLUTION: A river is flowing at a rate of 5 km/h. A boat travels 12km upstream and 36km downstream in a total of 9hours. What is the speed of the boat in still water?

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Question 161585: A river is flowing at a rate of 5 km/h. A boat travels 12km upstream and 36km downstream in a total of 9hours. What is the speed of the boat in still water?
Found 2 solutions by KnightOwlTutor, Alan3354:
Answer by KnightOwlTutor(293) About Me  (Show Source):
You can put this solution on YOUR website!
We know that the ratio of the distance travel is 3:1 36:12
We then make the distance equal by dividing the distance traveled downstream by 3
We know the speed of the river=5 km/h
x=time it takes to travel upstream
9-x time it takes to travel downstream
5x=1/3(9-x)(5)
This equation is based on D=(rate)(time)
Muliply both sides by 3
15x=(9-x)(5)
Use the distributive property
15x=45-5x
add 5x to both sides
20x=45
divide both sides by 20 x=2.25 9-x=6.75
Lets check the ratio to make sure that it is 3:1
6.75/2.25=3
Let's plug into the D=TR to calculate rate
36/6.75=5.33 km/hr
12/2.25=5.33 km/hr
The rate is the same.

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
A river is flowing at a rate of 5 km/h. A boat travels 12km upstream and 36km downstream in a total of 9hours. What is the speed of the boat in still water?
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B = speed of boat
B-5 = speed of boat upstream
B+5 = speed downstream
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h = hours going upstream
h*(B-5) = 12
(9-h)*(B+5) = 36
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B = 12/h + 5
B = 36/(9-h) - 5
B = 12/h + 5 = 36/(9-h) - 5
12/h + 10 = 36/(9-h)
Multiply by h*(9-h)
12(9-h) + 10h(9-h) = 36h
108-12h + 90h-10h^2 = 36h
Collect terms
-10h^2 + 42h + 108 = 0
5h^2 - 21h - 54 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 5x%5E2%2B-21x%2B-54+=+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=%28-21%29%5E2-4%2A5%2A-54=1521.

Discriminant d=1521 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--21%2B-sqrt%28+1521+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-21%29%2Bsqrt%28+1521+%29%29%2F2%5C5+=+6
x%5B2%5D+=+%28-%28-21%29-sqrt%28+1521+%29%29%2F2%5C5+=+-1.8

Quadratic expression 5x%5E2%2B-21x%2B-54 can be factored:
5x%5E2%2B-21x%2B-54+=+%28x-6%29%2A%28x--1.8%29
Again, the answer is: 6, -1.8. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+5%2Ax%5E2%2B-21%2Ax%2B-54+%29

The online solver always uses x, so sub h for x.
The -1.8 hours is not usable, so the time going upstream is,
h = 6 hours.
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Sub h into B = 12/h + 5 to solve for B
B = 12/6 + 5
B = 7 kph
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The people should have gotten out and walked, woulda been faster.