SOLUTION: The Hudson River flows at a rate of 3mph. A patrol boat travels 60 miles upriver and returns in a total time of 9 hr. What is the speed of the boat in still water? D= rt upriv

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Question 113995This question is from textbook Introductory and Intermediate Algebra
: The Hudson River flows at a rate of 3mph. A patrol boat travels 60 miles upriver and returns in a total time of 9 hr. What is the speed of the boat in still water?
D= rt upriver - 9r = 60 still water - 3rt = 60
Have I set this equation up correctly? Where do I go from here? Please help me. This question is from textbook Introductory and Intermediate Algebra

Found 2 solutions by jim_thompson5910, checkley71:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let r=speed of the boat in still water

When the boat goes upstream, we subtract 3mph from r to get (the boat is slowed by 3mph)

60=%28r-3%29t%5B1%5D

When the boat goes downstream, we can add 3mph to r to get (the boat is sped up by 3mph)

60=%28r%2B3%29t%5B2%5D

Now since the distance is the same we can set the two equations equal to one another to get

60%2F%28r-3%29=t%5B1%5D Now solve for t%5B1%5D in the first equation


60%2F%28r%2B3%29=t%5B2%5D Now solve for t%5B2%5D in the second equation


Since the total time was 9 hr, this means t%5B1%5D%2Bt%5B2%5D=9


60%2F%28r-3%29%2B60%2F%28r%2B3%29=9 Plug in t%5B1%5D=60%2F%28r-3%29 and t%5B1%5D=60%2F%28r%2B3%29


%28r-3%29%28r%2B3%29%2860%2F%28r-3%29%2B60%2F%28r%2B3%29%29=%28r-3%29%28r%2B3%29%289%29 multiply by the LCD %28r-3%29%28r%2B3%29


60%28r%2B3%29%2B60%28r-3%29=%28r-3%29%28r%2B3%29%289%29 Distribute


60%28r%2B3%29%2B60%28r-3%29=%28r%5E2-9%29%289%29 Foil


60r%2B180%2B60r-180=9r%5E2-81 Distribute again


120r=9r%5E2-81 Combine like terms


0=9r%5E2-81-120r Subtract 120r from both sides


0=9r%5E2-120r-81 Rearrange the terms


Now let's use the quadratic formula to solve for r:


Starting with the general quadratic

ar%5E2%2Bbr%2Bc=0

the general solution using the quadratic equation is:

r+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29



So lets solve 9%2Ar%5E2-120%2Ar-81=0 ( notice a=9, b=-120, and c=-81)




r+=+%28--120+%2B-+sqrt%28+%28-120%29%5E2-4%2A9%2A-81+%29%29%2F%282%2A9%29 Plug in a=9, b=-120, and c=-81



r+=+%28120+%2B-+sqrt%28+%28-120%29%5E2-4%2A9%2A-81+%29%29%2F%282%2A9%29 Negate -120 to get 120



r+=+%28120+%2B-+sqrt%28+14400-4%2A9%2A-81+%29%29%2F%282%2A9%29 Square -120 to get 14400 (note: remember when you square -120, you must square the negative as well. This is because %28-120%29%5E2=-120%2A-120=14400.)



r+=+%28120+%2B-+sqrt%28+14400%2B2916+%29%29%2F%282%2A9%29 Multiply -4%2A-81%2A9 to get 2916



r+=+%28120+%2B-+sqrt%28+17316+%29%29%2F%282%2A9%29 Combine like terms in the radicand (everything under the square root)



r+=+%28120+%2B-+6%2Asqrt%28481%29%29%2F%282%2A9%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



r+=+%28120+%2B-+6%2Asqrt%28481%29%29%2F18 Multiply 2 and 9 to get 18

So now the expression breaks down into two parts

r+=+%28120+%2B+6%2Asqrt%28481%29%29%2F18 or r+=+%28120+-+6%2Asqrt%28481%29%29%2F18


Now break up the fraction


r=%2B120%2F18%2B6%2Asqrt%28481%29%2F18 or r=%2B120%2F18-6%2Asqrt%28481%29%2F18


Simplify


r=20+%2F+3%2Bsqrt%28481%29%2F3 or r=20+%2F+3-sqrt%28481%29%2F3


So these expressions approximate to

r=13.9772373998204 or r=-0.643904066487102


So our possible solutions are:
r=13.9772373998204 or r=-0.643904066487102

However, we cannot have a negative speed, so our only solution is r=13.9772373998204


So the speed of the boat in still water is about 13.977 mph

Answer by checkley71(8403) About Me  (Show Source):
You can put this solution on YOUR website!
D=RT OR T=D/R
60/(R-3)+60/(R+3)=9
[60(R+3)+60(R-3)]/(R-3)(R+3)=9
(60R+180+60R-180)/(R^2-9)=9 (THE +180 & THE -180 CANCEL OUT)
120R/(R^2-9)=9 NOW CROSS MULTIPLY
9R^2-81=120R
9R^-120R-81=0
USING THE QUADRATIC EQUATION WE GET:
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
R=(120+-SQRT[120^2-4*9*-81])/2*9
R=(120+-SQRT[14,400+2916])/18
R=(120+-SQRT17,316)/18
R=(120+-131.59)/18
R=(120+131.59)/18
R=251.59/18
R=13.977 MPH IS THE SPEED OF THE BOAT IN STILL WATER.