From: Fred Bloggs
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Subject: Re: Complex number quiz
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Date: Wed, 11 Sep 2002 13:14:03 GMT
NNTP-Posting-Date: Wed, 11 Sep 2002 06:14:03 PDT
Organization: EarthLink Inc. -- http://www.EarthLink.net
Jonathan Kirwan wrote:
> On Tue, 10 Sep 2002 16:54:50 GMT, Fred Bloggs
> >About 35.25 meters or so with a hand calculator zero-solving. There is
> >nothing profound here. You add up some arc areas and one triangle area
> >and you end up with an equation of the form:
> >8( x^2+0.25-x*sin(theta))+ psi/2-4*x*sin(theta)=pi
> >where cos(theta)=x and cos(psi)=1-2*x^2
> >and x=R/D where R=length of rope and D= diameter of circle.
> >Solve for x.
> >I may have made mistake but this is essentially it.
> Well, I can't tell about the essence since I don't follow your logic
> above. But if you draw it out, you'll see that 35.25 is far enough
> off to be obviously wrong, just by inspection. It's not off by just a
> little difference in area. Your answer gobbles up almost 68% of the
> area. Your answer does yield an angle, from the tethering point to
> the two intersections where the tethered cow reaches the edge of
> field's circle, which is about 90.34 degrees, though. No idea if that
> means anything to you. It doesn't, to me.
> I'll post the correct logic, elsewhere. But the numerical answer is
> that the tether should be 28.968211826m, approximately.
Okay- thanks. If you look at my diagram which shows the tether stretched
to its furthest intersection along the circumference for the upper half
of the circle, then the area available to the cow is simply the arc area
formed by R, which is 0.5*R^2*theta, plus the arc area formed by psi,
which is 0.5*D/2*psi, minus the isosceles R-D/2-D/2 triangle area
counted twice, which is 0.5*R*D/2*sin(theta). This algebraic sum is
equated to 1/4 the circle area, function of D= diameter. This is all
very simple, self-evident, and does not require calculus. The angles are
computed using the cosine formula e.g. R^2=2*(D/2)^2-2(D/2)^2*cos(psi)
etc. My numeric answer is for the much tougher case of when the
herbivore is forbidden to eat the grass from area inside the triangle-)
> I'll post the correct logic, elsewhere.
Right- well compare the above high school level model to your method
which nearly brings in Stokes theorem. This is a straightforward little
non-problem misrepresented as something tougher because no "solution"
has ever been found on the internet.