Question 165125: 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 gonzo(654)   (Show Source):
You can put this solution on YOUR website! assume the river is flowing downstream at 3mph.
let b-3 = rate of boat going upstream.
let b+3 = rate of boat going downstream.
let x = number of hours it took to go upstream.
let y = number of hours it took to go downstream.
total hours to go upstream and downstream = 9
total distance traveled = 120 (60 up and 60 down)
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total hours = 9
x+y = 9 (first equation) *****************************
total distance traveled = 120
rate of the boat going upstream at b-3 mph for x hours = 60
rate of the boat going downstream at b+3 mph for y hours = 60
(x*(b-3))+(y*(b+3)) = 120 (second equation) ****************************
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you have 2 equations in 3 unknowns.
this makes it difficult since you can't solve directly.
first you have to establish a relationship between 2 of the unknowns and then use that relationship to solve for one of those unknowns.
once you have solved for one of the unknowns, then your equation becomes 2 equations in 2 unknowns which can be solved directly.
if there's an easier way i don't know it.
here's what i did.
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first i developed a relationship between y and b as follows.
if x + y = 9, then x = 9-y.
first relationship is:
x = 9-y
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then i took the second equation and solved for y in relationship to b.
(x*(b-3))+(y*(b+3)) = 120
remove parentheses:
x*b - x*3 + y*b + y*3 = 120
substitute 9-y for x:
(9-y)*b - (9-y)*3 + y*b + y*3 = 120
remove parentheses:
9*b - y*b - 9*3 - (-y*3) + y*b + y*3 = 120
simplify:
9*b - y*b - 27 + y*3 + y*b + y*3 = 120
9*b - 27 + y*6 = 120 (y*b and -y*b cancel out)
add 27 to both sides:
9*b + y*6 = 147
subtract 9*b from both sides:
y*6 = 147 - 9*b
divide both sides by 6:
y = (147-9*b)/6
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take the half of the trip that uses y and solve for b.
y*(b+3) = 60
i have a relationship for y in terms of b so i can solve for b.
this equation becomes
substituting (147-9*b)/6 for y, and the equation becomes:
((147-9*b)/6)*(b+3) = 60
multiply both sides of the equation by 6:
(147-9*b)*(b+3) = 60*6
remove parentheses by multiplying out.
147*b + 3*147 -9*b^2 -27*b = 60*6
simplify:
147*b + 441 - 9*b^2 - 27*b = 360
combine like terms:
120*b + 441 - 9*b^2 = 360
add 9*b^2 to both sides of equation:
120*b + 441 = 9*b^2 + 360
subtract 120*b from both sides of the equation and subtract 360 from both sides of the equation:
441 - 360 = 9*b^2 - 120*b
simplify:
81 = 9*b^2 - 120*b
subtract 81 from both sides of the equation:
0 = 9*b^2 - 120*b - 81
which is the same as:
9*b^2 - 120*b - 81 = 0
divide both sides by 3:
3*b^2 - 40*b - 27 = 0
using the quadratic formula i was able to determine the value of b.
b = 13.9772... (full value stored in calculator). ************************
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now that i know b, the equation should be able to be solved.
we still have x + y = 9 (first equation)
we still have x(b-3) + y(b+3) = 120 (second equation)
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we can solve for x as follows:
x = 9-y
second equation becomes:
(9-y)*(b-3) + y(b+3) = 120
remove parentheses:
9*b - 27 -y*b + y*3 + y*b + y*3 = 120
simplifying:
9*b - 27 + y*6 = 120 (-y*b and +y*b cancel out)
add 27 to both sides:
9*b + y*6 = 147
subtract 9*b from both sides:
y*6 = 147 - 9*b
divide both sides by 6:
y = (147-9*b)/6
since b = 13.9772..., equation becomes:
y = (147-9*13.9772...)/6
simplifying:
y = 3.5341... (full value stored in calculator). **************************
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now that we have y and b, we can solve for x.
since x + y = 9, then
x = 9-y
substituting 3.5341... for y, and x becomes
x = 5.4658... (full value stored in calculator). ****************************
we now have solved for all of the unknowns and need to check to see if the answer is correct.
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answer is:
b = 13.9772....
y = 3.5341....
x = 5.4658....
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x + y = 9 already proven to be correct since it was used to find x.
second equation is:
(x*(b-3))+(y*(b+3)) = 120
substituting values above for b, y, and x makes equation become:
5.4658...*(10.9772...) + 3.5341...*(16.9772...) = 120
multiplying out using values stored in calculator:
120 = 120
which checks out.
answer is proven correct and it is (using most digits stored in computer):
b = 13.9772374 = speed of the boat.
y = 3.5341439 = number of hours it took to get downstream (with the current).
x = 5.4658561 = number of hours it took to get upstream (against the current).
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you should be able to use these numbers to prove it yourself.
answer will be either very close or right on.
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gonzo@gmx.us
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