SOLUTION: The Hudson River flows at a rate of 3 mile per hour. A patrol boat travels 60 miles upriver, and returns in a total time of 9 hours. What is the speed of the boat in still water?

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Question 149899This question is from textbook
: The Hudson River flows at a rate of 3 mile per hour. A patrol boat travels 60 miles upriver, and returns in a total time of 9 hours. What is the speed of the boat in still water? This question is from textbook

Answer by nerdybill(7384) About Me  (Show Source):
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The Hudson River flows at a rate of 3 mile per hour. A patrol boat travels 60 miles upriver, and returns in a total time of 9 hours. What is the speed of the boat in still water?
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Let x = speed of patrol boat in still water
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Time it takes to go up river:
60/(x-3)
.
Time it takes to go down river:
60/(x+3)
.
Therefore, we have:
60/(x-3) + 60/(x+3) = 9
multiplying both sides by (x-3)(x+3) we get:
60(x+3) + 60(x-3) = 9(x-3)(x+3)
60x + 180 + 60x - 180 = 9(x^2-9)
120x = 9x^2 - 81
0 = 9x^2 - 120X - 81
0 = 3x^2 - 40X - 27
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Since we can't factor, we must use the quadratic equation. The solutions it gives is a positive and a negative answer. Since it can't be negative we use the positive one.
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x = 14 mph
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Quadratic solution follows:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 3x%5E2%2B-40x%2B-27+=+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=%28-40%29%5E2-4%2A3%2A-27=1924.

Discriminant d=1924 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--40%2B-sqrt%28+1924+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-40%29%2Bsqrt%28+1924+%29%29%2F2%5C3+=+13.9772373998204
x%5B2%5D+=+%28-%28-40%29-sqrt%28+1924+%29%29%2F2%5C3+=+-0.643904066487103

Quadratic expression 3x%5E2%2B-40x%2B-27 can be factored:
3x%5E2%2B-40x%2B-27+=+3%28x-13.9772373998204%29%2A%28x--0.643904066487103%29
Again, the answer is: 13.9772373998204, -0.643904066487103. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B-40%2Ax%2B-27+%29