KindofSlow
Junior Member
- Joined
- Mar 5, 2010
- Messages
- 90
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
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