Bob is having a nice camping/fishing trip along a river. He leaves his campsite early in the morning, and gets on his boat, heading full throttle upstream.

After going for exactly one mile, his hat flips off of his head, and starts floating downstream. Bob doesn't realize that his hat has fallen off for five minutes, but then he notices that it's missing, and turns full throttle downstream.

He finally catches the hat at exactly the same spot as he camped that morning. The question is, how fast was the water traveling?

(Assume that he travels the same speed the entire time and that there is no turn around time.)

There are only three facts given in the problem:
Bob had traveled exactly one mile when his hat flew off,
he turned around exactly five minutes later,
and he found the hat at the same spot (on the bank) he originated from.
Let x be the full-throttle speed of Bob's boat (relative to the water) in miles/minute.
Let y be the speed of the water downstream.

When Bob first starts, he is going full throttle upstream. Thus, the speed of his boat is going against that of the river, and his speed relative to the shore is (x-y) miles per minute.
He traveled the first mile, then, in 1/(x-y) minutes, and he traveled 5(x-y) miles in the next 5 minutes.

The return trip was, of course, exactly the same distance along the shore as he has already come, or 1+5(x-y) miles upstream. The time it took him was the same as the time it took his hat to float downstream (which is moving at the same speed as the river) one mile, which would be 1/y minutes. Finally, his speed going downstream, with the current now, is (x+y)mi/min.

So, we have one equation with two unknowns, which means that unless I missed something in the problem, it can be solved, but with multiple solutions.

The equation is rate=distance/time, or:
x+y = [1+5(x-y)]/[1/y]
which becomes:
x+y = y(1+5x-5y)
x+y = y+5xy-5y²
I don't know how to solve for a specific value of y, but to find one solution, just let y=1:
x+(1) = 1 + 5x(1) -5(1²)
5=4x
x=5/4. Or, if the full-throttle speed of his boat is 5 miles/4 minutes, then the water is traveling at 1 mile/minute.

Did I miss something?

Posted by DJ on 2003-05-05 10:08:46