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Calculus Mazimizing Problem

  1. May 5, 2012 #1
    A farmer can use one mile of a straight river as one border of an animal pen. What is the least amount of fencing needed to enclose a region of one square mile?

    I'm guessing it is a semi circular pen. But im not sure how to show that that is the minimum. I know its smaller than the triangle and the square.

    Any advice on how to go about this?
     
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  3. May 5, 2012 #2

    tiny-tim

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    Hi Punkyc7! :smile:

    Hint: can you make the problem more symmetric? :wink:
     
  4. May 5, 2012 #3
    what is more symmetric then a semi circle
     
  5. May 5, 2012 #4

    tiny-tim

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    exactly! :wink:
     
  6. May 5, 2012 #5
    Ok so it should be the semi circle.
     
  7. May 5, 2012 #6

    tiny-tim

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    maybe and maybe not

    you're thinking of making the solution more symmetric

    i'm suggesting making the problem more symmetric :wink:
     
  8. May 5, 2012 #7
    hmm... I dont quite follow what you're hinting at.
     
  9. May 5, 2012 #8

    tiny-tim

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    What is asymmetric about the problem?

    How would you get rid of the asymmetry? :wink:
     
  10. May 5, 2012 #9

    sharks

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    This is like one of those I.Q. test questions. :smile:
     
  11. May 5, 2012 #10
    divide by 2?
     
  12. May 6, 2012 #11

    sharks

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    Divide what by 2?
     
  13. May 6, 2012 #12

    sharks

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    Since the OP has not responded, i'm going to venture a guess... The least amount of fencing needed is 3 miles? :uhh:
     
  14. May 6, 2012 #13

    HallsofIvy

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    That has nothing to do with the question the OP originally asked or was asked to respond to. What is more symmetric than a semi-circle? A circle.

    In order that this be useful however do you know or have you proved that the figure containing a fixed area for the smallest perimeter (without any conditions) is a circle?
     
    Last edited: May 6, 2012
  15. May 6, 2012 #14
    I think that is a well know fact. But If I use a whole circle I am adding extra perimeter which is not what I want to do.... I cant seem to figure out how I would use calculus to solve this.
     
  16. May 6, 2012 #15

    sharks

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    I've tried to prove this by finding the perimeter of a circle with radius, r, and fixed area (a constant value, say, 4 miles square). Then, tested the same fixed area with square, equilateral triangle and isosceles triangle. The results agree. I guess it's one of those secrets of geometry.
     
    Last edited: May 6, 2012
  17. May 6, 2012 #16

    sharks

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    The replies above seem to suggest finding the perimeter of the circle... and i think i solved the problem. :smile:
     
    Last edited: May 6, 2012
  18. May 6, 2012 #17
    so you think 3 is the correct answer?
     
  19. May 6, 2012 #18

    sharks

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    No. I didn't take into account the fact that the circle has the least perimeter for a fixed area. I originally thought it was a square with one side = 1 mile, along the river. This is how i had (wrongly) calculated the required length of fencing to be the remaining 3 equal sides.
     
  20. May 6, 2012 #19

    tiny-tim

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    The asymmetry is the river … the pen is only on one side of it.

    Suppose the pen is allowed to be on both sides of the river, and to be two square miles …

    how does the answer to that relate to the answer to the original question?​

    (i assume this is the way Sharks answered it :wink:)
     
  21. May 9, 2012 #20

    sharks

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    OK, since the OP apparently gave up (and this question has been gnawing at me), i'm going to suggest my answer: [itex]2\sqrt{\pi}-1[/itex]. Is it correct?
     
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