Crossing a river with a current (optimization)

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Homework Help Overview

The problem involves a man attempting to cross a river that is 50 meters wide while dealing with a current flowing downstream at 2.4 m/s. He has a swimming speed of 3 m/s and a walking speed of 10 m/s. The objective is to determine the optimal angle at which he should swim to minimize the time taken to reach a person directly across the river.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the relationship between swimming and walking speeds, questioning the relevance of walking speed in the context of crossing the river. Some suggest that the current affects the swimmer's path, necessitating a walk to reach the destination after swimming. Others express confusion about the calculus involved in finding the optimal angle and the differentiation process required.

Discussion Status

The discussion is ongoing, with participants exploring various interpretations of the problem setup and the calculations involved. Some have attempted to derive equations for the time taken to cross the river and are seeking clarification on the calculus needed to minimize that time. There is no explicit consensus yet, but several participants are actively engaging with the mathematical aspects of the problem.

Contextual Notes

There is some ambiguity regarding the distances involved after crossing the river, with participants estimating how far the current will push the swimmer downstream. The original poster expresses difficulty with the calculus required for optimization, and there is a mention of using computational tools for assistance.

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 across + the distance from the other person divided by 10 but I don't 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
 
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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?
 
The current speed pushes him away from the person so when he's on shore he has to walk the remainder.
 
so how much shore is there? like 10m is shore 30m is river, 10m is shore?
 
Nono, imagine the river is a rectangle and the width is 50. So let's 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
 
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TheAstroMan said:
What is the best angle for him to swim at to have the fastest time to cross the river?
It says to find the most efficient route there but
TheAstroMan said:
Nono, imagine the river is a rectangle and the width is 50. So let's say he swims straight and the current pushes him down the shore 120m . So he has now crossed the river but still needs to walk the remaining 120.

Thats what the walking speed is for :p
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 can't be bothered
 
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
 
It was exactly 40m down cause 50÷3m/s is 16.666666666666666666666666666667 Times 2.4 = exactly 40
 
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
 

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