Question 120264: Question:
James paddled his boat upstream for one mile. He continued for
another 15 minutes, then turned around and paddled downstream,
arriving at his starting point in exactly 1 hour.How fast is the
current of the stream?
Found 2 solutions by ankor@dixie-net.com, Will-I-Am:
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! James paddled his boat upstream for one mile. He continued for
another 15 minutes, then turned around and paddled downstream,
arriving at his starting point in exactly 1 hour.How fast is the
current of the stream?
:
:
I am not real happy with what I came up with, but here it is:
:
Going over what you had:
Solution:This is what i did,I defined 2 variables
B = speed of boat in still water, in miles per hour
C = speed of current, in miles per hour
Distance Rate Time
Upstream 1 mile + 1/4hr(B-C) B-C 1/4hr<<<<
The problem here is we don't know total time of the upstream trip at this point
:
The distance both ways can be written as (1+.25(B-C))
:
On the return trip, this distance was covered in 1 hr so we can also say:
(1+.25(B-C)) mph, the speed can also be written (B+C)mph
:
A Speed equation:
(1+.25(B-C)) = (B+C)
1 + .25B - .25C = B + C
1 +.25B - B - .25C - C = 0
-.75B - 1.25C = -1
or mult by -1
.75B + 1.25C = +1
Get rid of the decimals, multiply by 4
3B + 5C = 4
:
But we have only 1 equation and two unknowns. I have been unable to generate
another equation from the information given.
:
Taking the above equation and graphing it, where y is the speed in still water,
and x is the speed of the current:
3y + 5x = 4
3y = -5x + 4
y =  x +  ; plotting this we have:
:
You can see the positive values are very limited,
:
If you had a still water speed of 1 mph, the current would be .2 mph.
this would work in our problem, B=1; C=.2
:
Distance 1 + .25(1-.2) = 1 + .25(.8) = 1.2 mi is the distance
:
The downstream speed 1 + .2 = 1.2 mph and of course to cover 1.2 mi would be 1 hr
This would be the only still water speed that is an integer
:
As I said I am not real happy with it.
Answer by Will-I-Am(1)  (Show Source):
You can put this solution on YOUR website! If the first mile is >= 45 min (3/4 hr.), the upstream trip will be >= 1 hr., leaving no time for the downstream trip. The minimum upstream speed is therefore
(1 mi/0.75 hr) = 4/3 mph. (Minimum upstream speed)
The downstream speed cannot be less than the upstream speed, so the maximum downstream time is 30 min. This would be equal to the upstream time of 30 min, allowing 15 min for the first mile, or 4 mph.
4 mph (Maximum downstream speed)
Let r be the speed of the boat and s be the speed of the stream. Upstream speed is r-s, downstream is r+s. From the above discussion,
r - s >= 4/3 and
r + s <= 4
or
r >= s + 4/3
r <= -s + 4
We also know that s >= 0.
Graphing, the regions overlap in a triangular region with vertices at (s,r) =
(0,4/3)
(0,4)
(4/3, 8/3)
Possible Up- and downstream velocities are respectively
4/3, 4/3 (15 minutes upstream gives 1/3 mi)
4, 4 (15 min upstream gives 1 mi)
4/3, 4 (15 min upstream gives 1/3 mi)
Up- and downstream times are
3/4 + 1/4, 3/4 + 1/4 Total time = 2 hours. Contradiction.
1/4 + 1/4, 1/4 + 1/4 Total time = 1 hour. OK.
3/4 + 1/4, 4/3 + 1/4 Total time = 2 hrs. 35 min. Contradiction.
The stream is not moving; the boat moves at 4 mph.
You can also simplify the problem initially--ignore the speed of the stream by assuming that it is zero. In this case, the boat travels upstream and downstream at the same speed. For the round trip to take 1 hr, it would be 0.5 hr upstream and 0.5 hr downstream. The first mile would therefore take 15 min, the second part of the upstream trip would be 1 mi, and the entire trip would be 4 mi at a speed of 4 mph. This works, and any other speed would result in a round trip either more or less than 1 hr.
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