1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Circular pasture

  1. Nov 5, 2009 #1
    My maths teacher recently gave me a weird question which really got my attention, but I have not managed to solve it so far. It can't exactly be described as homework, although it would give me a huge bonus if I managed to hand in the results.


    There is a pasture in the shape of a circle. A cow is tied to a point on the circle. How long does the rope have to be, so the cow manages to pasture exactly 1/2 of the grassland?


    I'm looking forward to everybody's opinions, and thanks in advance!


    David
     
  2. jcsd
  3. Nov 5, 2009 #2
    To start, assume the pasture has radius R, then the area of the pasture is [tex]\pi R^2[/tex].

    It may help to assume that the center of the pasture lies on the positive x-axis, and that edge of the pasture passes through the origin.

    Now, assume that the cow's rope has radius C and is tied to the origin. This is the center of a second circle having area [tex]\pi C^2[/tex].

    You are looking for the intersection of these two circles.

    Note that you can assume C>R if it is to cover half of the pasture.
     
  4. Nov 5, 2009 #3
    Also, you know that [tex]C < R\sqrt{2}[/tex], since having [tex]C = R\sqrt{2}[/tex] would mean that circle C (the cow's circle) would intersect circle R (the pasture circle) at (R,R) and (R,-R). This intersection is obviously greater than 1/2 the area of circle R.

    So now we have [tex]R < C < R\sqrt{2}[/tex].
     
  5. Nov 5, 2009 #4
    Assume the intersecting points between the two circles are A (at the top) and B (at the bottom). Line segment AB is a chord of both circles, and thus we have 2 circle segments.

    This should give you enough information to get you on the right track...
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Circular pasture
  1. Circular arc (Replies: 5)

Loading...