Results 1 to 4 of 4

Math Help - grazing goat

  1. #1
    Eater of Worlds
    galactus's Avatar
    Joined
    Jul 2006
    From
    Chaneysville, PA
    Posts
    3,001
    Thanks
    1

    grazing goat

    This problem may be a little cliche, but I will post it anyway for those who have not seen it.

    A goat is tied to a silo of radius 12 feet by a tether 36 feet long. Calculate the area the goat can graze.

    Here is a hackneyed diagram. It kind of resembles a limacon.
    Last edited by galactus; November 24th 2008 at 05:38 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    If I remember correctly it is a cardioid.

    I think Stuart called it
    "Clara the calculus cow"

    This problem made me crazy for about a week.

    A cool more advanced way to solve this is to use Greene's thorem.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Eater of Worlds
    galactus's Avatar
    Joined
    Jul 2006
    From
    Chaneysville, PA
    Posts
    3,001
    Thanks
    1
    The length of the tether from where the goat is tied to the end of its rope is 36-12{\theta}.

    That's because the rope wraps around the silo before it becomes tangent to the silo at that point.

    The total angle the goat grazes is 3 because 12{\theta}=36

    There is a small piece it can not reach because the semi-diamter of the silo is slightly longer than 36 feet. Therefore, we don't have to integrate to Pi, but slightly less than that.

    Limits of integration are {\theta}=[0,3].

    The semicircular region is easy \frac{36^{2}\pi}{2}=648{\pi}

    The region we are interested in is the second and fourth quadrants where the goat goes around the silo.

    2\int_{0}^{3}\frac{1}{2}(36-2{\theta})^{2}d{\theta}=1296

    So, the total area is 648{\pi}=1296=3331.75 \;\ ft^{2}

    I'd like to see the Green's theorem method. I will look into it.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    I will work a variation of the problem.

    Suppose that a cow is tied to a silo of radius r with a rope of length \pi r

    See the diagram below

    grazing goat-silo.jpg

    using trig and the fact that the rope is always at a right angle to the silo we can find d_1,d_2,d_3,d_4 and parametric equations of the curve.

    Note the length of the hypotenuse of the small triangle is r\theta
    becuase it have the same length as the arc of the circle subtended by the angle theta.


    d_1=r\cos(\theta)
    d_1=r\theta\sin(\theta)

    so x as a function of theta is x=d_1+d_2 so

    x(\theta)=r(\cos(\theta+\theta\sin(\theta))

    using a similar argument for y we get

     <br />
y(\theta)=r(\sin(\theta)-\theta\cos(\theta))<br />

    if we graph this from -\pi \mbox{ to } \pi we get
    Note for the graph I will use r=12
    grazing goat-graph1.jpg

    Now for the other half of the equation we want the half circle
    centered at (-r,0) with a radius of \pi r

    x(\theta)=-r +\pi r \cos(theta)
    y(\theta)=\pi r \sin(theta)

    graphed in green theta \frac{\pi}{2}\mbox{ to } \frac{3\pi}{2}

    grazing goat-graph2.jpg

    These two curves enclose the total area for the cow, but this is too much area we need to remove the silo.

    grazing goat-graph3.jpg

    lets add one more line
    grazing goat-graph4.jpg

    let c_1 be the blue curve
    let c_2 be the purple curve
    let c_3 be the green curve

    Now for Green's theorem

    \iint_A1dA=-\int_{\partial A}ydx

    -\int_{c_1 \cup c_2}ydx=-\int_{c_1}ydx-\int_{c_2}ydx

    but dx=0 on the purple curve so we get...

    -\int_{-pi}^{pi}r(\sin(\theta)-\theta\cos(\theta))r\theta\cos(theta)d\theta=\frac  {1}{3}r^2\pi^3+r^2\pi

    The area of the other half circle is \frac{1}{2}r^2\pi^3

    Then we subtact off the area of the silo to get

    A=\frac{5}{6}\pi^3r^2

    whew that was longer than I remember.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Area of Goat Grazing
    Posted in the Geometry Forum
    Replies: 1
    Last Post: June 11th 2010, 10:06 AM
  2. Monty Hall goes Car-Goat-Key
    Posted in the Math Puzzles Forum
    Replies: 4
    Last Post: October 4th 2009, 05:09 PM
  3. Geometry - The area the goat can graze?
    Posted in the Geometry Forum
    Replies: 5
    Last Post: July 27th 2009, 07:57 AM

Search Tags


/mathhelpforum @mathhelpforum