1. Standard memberPBE6
    Bananarama
    False berry
    Joined
    14 Feb '04
    Moves
    28719
    19 Nov '04 19:56
    OK, here's a problem I'd like an analytical solution to:

    A stupid farmer, prone to idiocy, buys a circular plot of land. Dunce. Anyway, he has one (count 'em - 1!!!) sheep that he feed by allowing it to graze on the field. Unfortunately, the farmer also like to graze on the field because he's too stupid to realize that (a) he can milk the sheep and get some solid good cheese; and (b) he can kill the sheep afterwards and make a nice mutton, cheese and grass sandwich on two pieces of hand. So the farmer is only willing to let the sheep graze on 1/2 of the field.

    The farmer has one rope, which he might hang himself with if he can't solve this problem, but which he intends to tether the sheep with. Just 'cause he like to do things the hard way, the sheep will be tethered to the edge of the field, and a circular fence set up along the perimieter to keep the sheep from grazing on Farmer Sheisse's field across the way.

    The question is, how long should the rope be? Report the answer as a ratio of rope length to field diameter.
  2. Zeist, Holland
    Joined
    11 Sep '03
    Moves
    19337
    19 Nov '04 21:301 edit
    I tried putting this in Maple (use two functions for the circles (land and sheep-coverage), calculate the integral of the difference-function, etc). You wouldn't believe the load of crap I got. Maybe I did something wrong, maybe not. Have you calculated this for yourself?
  3. Standard memberPBE6
    Bananarama
    False berry
    Joined
    14 Feb '04
    Moves
    28719
    19 Nov '04 21:32
    I've done a numerical solution, and I've come up with an intergral expression, but I don't think I ever solved for the ratio of rope length to field diameter explicitly.

    F numerical solutions. F em to hell. I'd like to see something analytical.
  4. Joined
    26 Apr '03
    Moves
    25836
    19 Nov '04 23:24
    Originally posted by PBE6
    I've done a numerical solution, and I've come up with an intergral expression, but I don't think I ever solved for the ratio of rope length to field diameter explicitly.

    F numerical solutions. F em to hell. I'd like to see something analytical.
    I think I can get an expression without integrals, involving areas of chords of circles - but it is going to be hard if not impossible to solve it - have to wait till I'm fresh.
  5. Joined
    15 Oct '04
    Moves
    1995
    23 Nov '04 15:53
    rope should be .77475294626251 * the diameter of the field
  6. Standard memberPBE6
    Bananarama
    False berry
    Joined
    14 Feb '04
    Moves
    28719
    23 Nov '04 16:03
    Close, but no goddamn cigar. 3/4 of the field diameter is too big.
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