SOLUTION: 4. The Hudson River flows at a rate of 3 miles 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 wa

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: 4. The Hudson River flows at a rate of 3 miles 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 wa      Log On


   

Question 155379: 4. The Hudson River flows at a rate of 3 miles 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?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
# 4

Let x = speed of boat in still water



d=rt Start with the distance-rate-time formula



60=%28x-3%29t Plug in d=60 and r=x-3. This equation represents the upstream journey


60%2F%28x-3%29=t Divide both sides by x-3 to isolate "t"


So the expression for the time it takes to go upstream can be represented by the expression 60%2F%28x-3%29

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60=%28x%2B3%29t Plug in d=60 and r=x%2B3. This equation represents the downstream journey


60%2F%28x%2B3%29=t Divide both sides by x%2B3 to isolate "t"


So the expression for the time it takes to go downstream can be represented by the expression 60%2F%28x%2B3%29


Now simply add the two time expressions to get: 60%2F%28x-3%29%2B60%2F%28x%2B3%29


60%2F%28x-3%29%2B60%2F%28x%2B3%29=9 Now set that expression equal to the total time of 9 hours


60%28x%2B3%29%2B60%28x-3%29=9%28x%2B3%29%28x-3%29 Multiply every term by the LCD %28x%2B3%29%28x-3%29 to clear the denominators


60%28x%2B3%29%2B60%28x-3%29=9%28x%5E2-9%29 FOIL


60x%2B180%2B60x-180=9x%5E2-81 Distribute


60x%2B180%2B60x-180-9x%5E2%2B81=0 Subtract 9x%5E2 from both sides. Add 81 to both sides.


-9x%5E2%2B120x%2B81=0 Combine like terms


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=-9, b=120, and c=81


Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%28120%29+%2B-+sqrt%28+%28120%29%5E2-4%28-9%29%2881%29+%29%29%2F%282%28-9%29%29 Plug in a=-9, b=120, and c=81


x+=+%28-120+%2B-+sqrt%28+14400-4%28-9%29%2881%29+%29%29%2F%282%28-9%29%29 Square 120 to get 14400.


x+=+%28-120+%2B-+sqrt%28+14400--2916+%29%29%2F%282%28-9%29%29 Multiply 4%28-9%29%2881%29 to get -2916


x+=+%28-120+%2B-+sqrt%28+14400%2B2916+%29%29%2F%282%28-9%29%29 Rewrite sqrt%2814400--2916%29 as sqrt%2814400%2B2916%29


x+=+%28-120+%2B-+sqrt%28+17316+%29%29%2F%282%28-9%29%29 Add 14400 to 2916 to get 17316


x+=+%28-120+%2B-+sqrt%28+17316+%29%29%2F%28-18%29 Multiply 2 and -9 to get -18.


x+=+%28-120+%2B-+6%2Asqrt%28481%29%29%2F%28-18%29 Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


x+=+%28-120%2B6%2Asqrt%28481%29%29%2F%28-18%29 or x+=+%28-120-6%2Asqrt%28481%29%29%2F%28-18%29 Break up the expression.


So the answers are x+=+%28-120%2B6%2Asqrt%28481%29%29%2F%28-18%29 or x+=+%28-120-6%2Asqrt%28481%29%29%2F%28-18%29


which approximate to x=-0.644 or x=13.977


Since a negative speed doesn't make sense in this problem, this means that the only solution is x=13.977

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Answer:

So the speed of the boat in still water is approximately 13.98 mph (rounded to the nearest hundredth).