SOLUTION: Hello, I have been working on this problem for a while and can;t seem to come up with a reasonable solution. The Hudson River flows at a rate of 5 miles per hour. A patrol b

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Question 194736: Hello,
I have been working on this problem for a while and can;t seem to come up with a reasonable solution.
The Hudson River flows at a rate of 5 miles per hour. A patrol boat travels 40 miles upriver, and returns in a total time of 6 hours. What is the speed of the boat in still water?
D=R x T
r=40/6
r-5=40/6
40/r-5 + 40/r+5=6
r^2-25(40/r-5) + r^-25(40/r+5=6
40r^2+1000/r^2-25 + 40r^2-1000/r^2-25=6
(r-5)(r+5) 40/r-5 and (r-5)(r+5) 40/r+5
after cancelling: 40r+5 +40r-5=6
80r=6
r=74
I know that this is not correct, but I can't figure out where I went wrong.
Thanks,
Dave
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!

You got off on the wrong foot from the git-go. You said , but that can't be true because 40 is the one-way distance and 6 hours is the total time for both directions. Not to mention that you then said and both of those statements cannot be simultaneously true.

Let's begin anew.

Let r represent the speed of the boat in still water. Then r - 5 is the speed of the boat upriver (against the current) and r + 5 is the speed downriver. Let t represent the amount of time it took to go upriver, then 6 - t is the time it took to go downriver.

For the upriver trip:



and for the downriver trip:



Solving both equations for t we get:



and



Now that we have two things equal to t, we can set them equal to each other:



LCD is , so:



Expanding binomials, removing parentheses, collecting terms:



Now, you can either multiply both sides by or realize that if you have two equal fractions with identical denominators, then the numerators have to be equal as well to write:



Put in standard form:



Fortunately, this little puppy factors rather neatly:




Exclude the negative root because we are reasonably certain that the boat wasn't going backwards as a negative rate through the water would suggest, and the rate in still water is 15 mph.

Check:

A 40 mile upriver trip would take 40 divided by (15 - 5) = 4 hours. A 40 mile downriver trip would take 40 divided by (15 + 5) = 2 hours. Total time: 6 hours, answer checks.

John