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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?: 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(9162) About Me  (Show Source):
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# 4

Let x = speed of boat in still water



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



60=(x-3)t Plug in d=60 and r=x-3. This equation represents the upstream journey


60/(x-3)=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/(x-3)

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60=(x+3)t Plug in d=60 and r=x+3. This equation represents the downstream journey


60/(x+3)=t Divide both sides by x+3 to isolate "t"


So the expression for the time it takes to go downstream can be represented by the expression 60/(x+3)


Now simply add the two time expressions to get: 60/(x-3)+60/(x+3)


60/(x-3)+60/(x+3)=9 Now set that expression equal to the total time of 9 hours


60(x+3)+60(x-3)=9(x+3)(x-3) Multiply every term by the LCD (x+3)(x-3) to clear the denominators


60(x+3)+60(x-3)=9(x^2-9) FOIL


60x+180+60x-180=9x^2-81 Distribute


60x+180+60x-180-9x^2+81=0 Subtract 9x^2 from both sides. Add 81 to both sides.


-9x^2+120x+81=0 Combine like terms


Notice we have a quadratic equation in the form of ax^2+bx+c where a=-9, b=120, and c=81


Let's use the quadratic formula to solve for x


x = (-b +- sqrt( b^2-4ac ))/(2a) Start with the quadratic formula


x = (-(120) +- sqrt( (120)^2-4(-9)(81) ))/(2(-9)) Plug in a=-9, b=120, and c=81


x = (-120 +- sqrt( 14400-4(-9)(81) ))/(2(-9)) Square 120 to get 14400.


x = (-120 +- sqrt( 14400--2916 ))/(2(-9)) Multiply 4(-9)(81) to get -2916


x = (-120 +- sqrt( 14400+2916 ))/(2(-9)) Rewrite sqrt(14400--2916) as sqrt(14400+2916)


x = (-120 +- sqrt( 17316 ))/(2(-9)) Add 14400 to 2916 to get 17316


x = (-120 +- sqrt( 17316 ))/(-18) Multiply 2 and -9 to get -18.


x = (-120 +- 6*sqrt(481))/(-18) Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


x = (-120+6*sqrt(481))/(-18) or x = (-120-6*sqrt(481))/(-18) Break up the expression.


So the answers are x = (-120+6*sqrt(481))/(-18) or x = (-120-6*sqrt(481))/(-18)


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).