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The Grazing Goat: CR4 Challenge (09/15/08)

Posted September 14, 2008 5:01 PM

This week's CR4 Challenge Question:

A goat is placed in a fenced-in circular field with radius r. It is tethered to a point on the perimeter of the field by a rope which allows it to graze as far as the radius R. What is the value of R to the nearest cm, if the goat can graze exactly ½ the circular field area, and r = 10m

Thanks to SlideRuler for this question!

And the Answer Is....

The grazing area is segment ABC + segment ABD

Therefore ½*r²*(2θ-sin(2θ))+½*R²*(2Φ-sin(2Φ))=½*π*r². ... eq(1)

also we have θ+Φ+Φ = π (isosceles Δ origin,C,A) ... eq(2)

and r*sin(θ) = R*sin(Φ) (both equal to ½*AB) ... eq(3)

Replacing 2Φ with (θ-π); and replacing R with r*sin(Φ)/sin(θ); and using the relationship sin²(Φ) = ½*(1-cos(2Φ)) = ½*(1-cos(π-θ)) = ½*(1+cos(θ)) in equation (1) gives

(π-θ-sin(θ))+(2θ-sin(2θ)-π)*(1+cos(θ)/(2*sin²(&#952;))=0 ... eq(4)

This has only one variable θ, and the root can be found using Newton's approximation. xn+1 = xn - yn/y'n

Using a starting value of θ=1 gives successively

θ =1 --> 1.2143 --> 1.2327 --> 1.2354 --> 1.2358 --> 1.2359

and the corresponding values of R by using eq(2) to get Φ and eq(3) to get R; (remembering r=10m) are

R= 10m --> 11.41m -->11.56m --> 11.58m --> 11.59m --> 11.59m

The answer to the given problem is therefore 11.59m

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#98
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Re: The Grazing Goat: CR4 Challenge (09/15/08)

09/26/2008 2:07 AM

...."cete" sounds like a small whale.

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#101
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Re: The Grazing Goat: CR4 Challenge (09/15/08)

09/26/2008 7:18 AM

The coloration is about right...

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#103
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Re: The Grazing Goat: CR4 Challenge (09/15/08)

09/26/2008 3:20 PM

How did this thread go from ruminants (goats) to rodents (squirrels) to aquatic mammals (whales)?

I wonder if we otter let this go on!

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#105
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Re: The Grazing Goat: CR4 Challenge (09/15/08)

09/26/2008 4:44 PM

I suspect we need to beaver-y careful about them otters (I still have my otter in the freezer, by the way...)!

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Re: The Grazing Goat: CR4 Challenge (09/15/08)

09/27/2008 2:21 AM

Strange but true.........

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Re: The Grazing Goat: CR4 Challenge (09/15/08)

09/29/2008 7:08 AM

Reminds me of an old recipe I once read for "bobcat stew". It started off: "First, skin, debone, and dice one bobcat into 1" cubes." Note no mention of actually obtaining said bobcat - I guess this used to be much more routine than it would be today, what with the current scarcity of bobcats at the meat market.

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Re: The Grazing Goat: CR4 Challenge (09/15/08)

09/29/2008 7:28 AM

Ah, just give it a few months and bushmeat won't just be found in Nairobi markets. I'm already perplexed by the fact that Indian restaurants describe dishes as 'Meat <whatever>'. It must be kinda hard adapting an essentially vegetarian diet to our Western carnivorous tastes. Poor Del probably thought it was a good idea to baste himself lounge in the Sicilian sun . I suspect a Bobcat would leave an aspiring Chef with one-inch fingers !

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Re: The Grazing Goat: CR4 Challenge (09/15/08)

09/25/2008 12:31 PM

The proper wording goes like this.

  • A single tiger is referred to as a tiger.
  • Two tigers are referred to as a pair of tigers.
  • Any more than two is referred to as a shitload.
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Re: The Grazing Goat: CR4 Challenge (09/15/08)

09/29/2008 2:56 AM

This answer is on the internet I did not post it while the puzzle was running but some might like to look at it.

Regards JD.

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Re: The Grazing Goat: CR4 Challenge (09/15/08)

09/29/2008 3:41 AM

PS the solution.

Regareds JD.

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Re: The Grazing Goat: CR4 Challenge (09/15/08)

09/29/2008 4:25 AM

GA from me, though the answer is not developed by you. You are neither tired or retired JD

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