# Homework Help: Crossing a river with a current (optimization)

1. Apr 9, 2015

### TheAstroMan

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1. A man is on one side of a river that is 50 m wide. He is trying to get to someone directly on the other side. There is a current flowing down the stream at 2.4 m/s. His swimming speed is 3 m/s and his walking speed is 10 m/s .

What is the best angle for him to swim at to have the fastest time to cross the river?

I've tried looking for this and I've only seen questions where the person is not directly across, and my math isn't very strong so I'm not able to use those to answer the one I have :(

Thanks!

2. 50/(3cos(x))+(2.4-3sin(x))*(50/(3cos(x)))/10

^^ I've gotten here, which is the time to get accross + the distance from the other person divided by 10 but I dont understand the calculus you have to use after this equation.

3. I just plugged that equation into wolfram alpha's minimum calculator and got two numbers. I used the one without a variable and assuming it was differentiated I assumed it was in radians. So I divided that by Pi / 180 to convert it into degrees and somehow I got the answer.

Can someone please explain the whole minimum / differentiating process? Thanks :D

Last edited: Apr 10, 2015
2. Apr 9, 2015

### GurfX

what does his walking speed have to do with anything.. its not like theirs a bridge to cross so he should be just swimming right?

3. Apr 9, 2015

### TheAstroMan

The current speed pushes him away from the person so when he's on shore he has to walk the remainder.

4. Apr 9, 2015

### GurfX

so how much shore is there? like 10m is shore 30m is river, 10m is shore?

5. Apr 9, 2015

### TheAstroMan

Nono, imagine the river is a rectangle and the width is 50. So lets say he swims straight and the current pushes him down the shore 40m . So he has now crossed the river but still needs to walk the remaining 120.

Thats what the walking speed is for :p

Last edited: Apr 9, 2015
6. Apr 9, 2015

### GurfX

It says to find the most efficient route there but
if walking the question would be easier so his walking speed is 10m/s which (distance to walk)120/10(meters per seccond) = 12 seconds to walk up to the destination

for your question it would take (50meters ÷ 3m/s = 16.66~ seconds to cross which in that time would bring him down the river 2.4 x 16.66 = 40m down the river (roughly) so then 4 seconds to walk up to the destination

My math isn't on edge but somebody should know how to find the angle to get to the end most efficiently but i cant be bothered

7. Apr 9, 2015

### TheAstroMan

Alright thanks. I messed up the math, its approximately 40m down not 120 haha oops.

But yea, I think I need someone who knows calculus for this :D

8. Apr 9, 2015

### GurfX

It was exactly 40m down cause 50÷3m/s is 16.666666666666666666666666666667 Times 2.4 = exactly 40

9. Apr 9, 2015

### TheAstroMan

I got here

50/(3cos(x))+(2.4-3sin(x))*(50/(3cos(x)))/10

^^ Basically the time it takes him to get across + his distance from the other person divided by 10. (I think this is the right equation, not sure though)

Now I just need someone to explain how I can get a value for x that would give me the smallest answer :p