Angle for fastest time across river

KindofSlow

Junior Member
Joined
Mar 5, 2010
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Problem - What angle to take to directly across river in minimal time.
Facts - river is 1,200 m wide, current = 0.8 m/s, boat = 1.6 m/s, and run along shore = 3.0 m/s.
My work
Time to get boat across river is 1,200/1.6*cos(x) = 750/cos(x)
Distance to run along river to reach point directly across from starting point =
1,200*tan(x) (this is distance if no current) - 600/cos(x) (this distance current pushes us downstream (750/cos(x))*0.8)
Time to run along river is this distance/3.00 m/s running velocity = 400*tan(x) - 200/cos(x)
So total time is 750/cos(x) + 400*tan(x) - 200/cos(x) = 550/cos(x) + 400*tan(x).
I want to find the minimum so I take the derivative and set equal to zero:
550*sec(x)*tan(x) + 400*sec^2(x) = 0
550*sin(x) + 400 = 0
sin(x) = -400/550
x = -46.7 degrees
Which is not right.
If someone could point out where my error is, I woud greatly appreciate it.
Thank you
 
Please define your variable. Is it an angle from the direct route or and angle from the present shore or something else entirely?
 
Last edited:
Problem is to find angle to depart from present shore.
Destination is at zero degrees/radians straight across river.
Make sense?
Thank you
 
1) Two parallel lines cut by a transversal. What do we know of the alternate interior angles?
2) If we are measuring from the present shore, wouldn't the point opposite be at pi/2, rather than zero (0)?
 
Here is the entire text of the problem in case my summary was too vague.

In hot pursuit, Agent Logan of the FBI must get directly across a 1,200 meter wide river in minimum time. The river’s current is 0.80 m/s, he can row a boat at 1.60 m/s, and he can run 3.00 m/s. Describe the path he should take (rowing plus running along the shore) for the minimum crossing time.

I've set up the problem with the river as vertical and the current flowing downward from north to south, and the racer on the left (west) shore so that his final destination is directly across the river on the right (east) shore.

Thank you
 
The problem remains in the clarity of your definitions. WHAT is 'x'? You have not stated it.

Try this. x = The heading of the boat. The direction we are TRYING to row. The direction we would go if there were no current.

x ranges from 0 (due north) to pi (due south). Of course, either of these endpoints would not be a rational solution, sinc ehte river would not be crossed.

Note: You MAY have overlooked that rowing could be pointed north, against the current.
 
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