SOLUTION: For a portion of the Green River in Utah, the rate of the river's current is 4 mph. A tour guide can row 6 mi down this river and back in 2 h. Find the rowing rate of the guide in

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Question 769071: For a portion of the Green River in Utah, the rate of the river's current is 4 mph. A tour guide can row 6 mi down this river and back in 2 h. Find the rowing rate of the guide in calm water.
I'm having a hard time recalling how o set up a problem like this. I have tried:
with current distance: 6mi
with current rate: x+4
with current time: 6/x+4
Against current distance: 6mi
against current rate: x-4
against current time: 6/x-4
I'm unsure of how to set this up exactly with all this data or if I am even on the right track. Thanks for any help.
Found 2 solutions by ramkikk66, josgarithmetic:
Answer by ramkikk66(644) About Me  (Show Source):
You can put this solution on YOUR website!


For a portion of the Green River in Utah, the rate of the river's current is 4 mph. A tour guide can row 6 mi down this river and back in 2 h. Find the rowing rate of the guide in calm water. 

I'm having a hard time recalling how o set up a problem like this. I have tried:

with current distance: 6mi

with current rate: x+4

with current time: 6/x+4 

Against current distance: 6mi

against current rate: x-4

against current time: 6/x-4 

Ans:

You have started on the right track. As you have calculated till now, if we

assume that x is the speed of the boat in still water.

Time for downstream = 6/(x+4)

Time for upstream = 6/(x-4)

It is given that the total time (up and down) is 2 hours. So we get the

equation

6%2F%28x%2B4%29+%2B+6%2F%28x-4%29+=+2 Cross multiplying

6%2A%28x-4%29%2B6%2A%28x%2B4%29+=+2%2A%28x%2B4%29%2A%28x-4%29 Expanding the terms,

6%2Ax+-+24+%2B+6%2Ax+%2B+24+=+2%2A%28x%5E2+-+16%29 

12%2Ax+=+2%2Ax%5E2+-+32 Simplifying

x%5E2+-+6%2Ax+-+16+=+0

This is a standard quadratic equation that you can solve by factorizing.

You get

x%5E2+-+8%2Ax+%2B+2%2Ax+-+16+=+0

x%2A%28x+-+8%29+%2B+2%2A%28x+-+8%29+=+0

%28x+-+8%29%2A%28x+%2B+2%29+=+0

x = 8 or x = -2

Since x cannot be negative, the speed of the boat in calm water is 8 mph.



Check for correctness:

If speed in still water is 8 mph

Time for downstream = 6/(8+4) = 0.5 hours

Time for upstream = 6/(8-4) = 1.5 hours

Total time = 0.5 + 1.5 = 2 hours.

Correct!



Hope you got it :)

Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Make a speed, time, distance table for all of the data and any that you can express from the given data.

Let r = speed of the guide if rowing in still water.
When you say, "6 mi down the river and back", this is 6 miles up the river and 6 miles down the river. Round trip up and back is 12 miles. If you believe this is the wrong interpretation then say so.


DIRECTION________speed___________time hours________distance miles
DOWN RIVER_______r+4_____________(___)________________6
UP RIVER_________r-4_____________(___)________________6

The key relation speed*time=distance allows us to make expressions to fill-in times.
r*h=d for speed%2AtimeHours=distanceMiles
timeHours=distanceMiles%2Fspeed


DIRECTION________speed___________time hours________distance miles
DOWN RIVER_______r+4_____________(6/(r+4))________________6
UP RIVER_________r-4_____________(6/(r-4))________________6
TOTAL______________________________2_____________________12

He rows up and back in 2 hours. This allows us to equate to the sum of the time expressions and the only unknown variable is r.
highlight%286%2F%28r%2B4%29%2B6%2F%28r-4%29=2%29 The problem's principle resulting equation to solve. The rest of the process is for you to do.