From Precalculus, Seventh Edition, Michael Sullivan:
A cow is tethered to one corner of a square barn, 10 feet by 10 feet, with a rope 100 feet long. What is the maximum grazing area for the cow? [Hint: See the illustration.]
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The key to this is to use a larger triangle that includes A3 and half the barn by extending a side of A3 to the SW corner of the barn.?ÿ That will get you two sides and a non-included angle. The rest is just plug and chug. Find the other side of the?ÿ new triangle, then side of A3, its area, and the supplementary angle of A2, etc.
Note that the drawing is not to scale and the corner o the circle shown is not centered at the corner of the barn.
Need more info.
How many times has it walked around the barn in the same direction?
@dmyhill?ÿ
That's a good point; what if she keeps walking, clockwise? She will lose a lot of ground in that SE corner...
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But if she takes advantage, of every square foot:
At first I thought someone had a video of me at a Mexican restaurant.?ÿ But now that I opened the thread...
I come up with 31,245.16 square feet.
I left the 100 square feet of the 10' x 10' barn in the equation.... this is why I have a second set of eyes on all of my plats.
A1+2*A2+2*A3 = 31,145.152 sq ft.
A1= 23561.945 ?ÿ?ÿ A2=3499.389 ?ÿ?ÿ A3=292.214
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This is of course the idealized math problem.?ÿ In practice there is the issue that animals wouldn't tend to utilize both directions of travel, and the cow may be able to reach slightly further or less than 100 ft if the rope is tied to a halter, even after allowing for the length of rope used up in knots at each end.
A1+2*A2+2*A3 = 31,145.152 sq ft.
A1= 23561.945 ?ÿ?ÿ A2=3499.389 ?ÿ?ÿ A3=292.214
?ÿ
This is of course the idealized math problem.?ÿ In practice there is the issue that animals wouldn't tend to utilize both directions of travel, and the cow may be able to reach slightly further or less than 100 ft if the rope is tied to a halter, even after allowing for the length of rope used up in knots at each end.
Also leading me to believe that cows with longer necks get more food. 🙂
I objected to the diagram, similar to Bill's comment. It seems that I find more and more things objectionable nowadays.
C1 is the rope tie point. As she turns the corner at C2, the radius of her circle shortens to 90 feet. Had she gone in the other direction, the same thing would have happened at C4. So, I make two equal sectors and a triangle that includes a diagonal of the barn as its base. Summary numbers are below. The red curves are Bessie's limits.
I just didn't like having two unequal sectors, but there is teaching value in having them. In either approach, the centers of the two small sectors are at different corners of the barn, but the corner pairs are different.?ÿ
The next problem on the page changes the dimensions of the barn to 10 x 20 and asks the same question.
In any event, if Bossie lives in New Mexico, she may be in trouble.
What if....ahhh nevermind.
What if....ahhh nevermind.
A 10' x 10' Barn? ?????ÿ
0.71 acres may keep her happy for a few days but not indefinitely.?ÿ Unless there is a mixture of Mary J plants in the grazing mix.?ÿ Then, she might last two days before eating all the forage available, the barn and the rope.
@gary_g?ÿ
A footnote to the problem:
"Suggested by Professor Teddy Koukounas of Suffolk Community College, who learned of it from an old farmer in Virginia. Solution provided by Professor Kathleen Miranda of SUNY at Old Westbury."
I wonder if anyone shared the solution with the old farmer? He probably knew that it was 3/4 of an acre, give or take.
It's a well-traveled problem; even been on Chegg.
0.71 acres may keep her happy for a few days but not indefinitely.?ÿ Unless there is a mixture of Mary J plants in the grazing mix.
Words of wisdom from the psilocybin mushroom King of Kansas. ?????ÿ
10x10 is a shed, not a barn.
The rope is tied to the cows neck so when she points her backside at the shed corner tie point she can reach a bit more than 100??.
it??s a dumb question.
But, is the rope 100' before any of it tied to the tethering point and the cows neck, so it could be conceivably shorter by feet, so maybe the cow can't even reach the 100' radius.
Now, everybody remember that this math is a deterministic model while we live in a?ÿprobabilistic world.
That's why rules of thumb sometimes give better answers than rigorous computations, why thousandths of an acre are ridiculous, why flat maps work, and why I've lived for 76 years.
And, yes, it does matter whether the cow is tied to a ring attached to the building or to a stake driven as close to the building as possible.
But not much.
Standards.?ÿ Standards allow us to know specific data to help such odd problems.?ÿ For example, say an eye-bolt is used as the tie-off spot on the barn.?ÿ It helps to know the dimensions of eye-bolts, so as to add that dimension to the calculation of the problem cited above.?ÿ Here is a bit of info on eye-bolts courtesy of the American National Standards Institute (ANSI)?ÿ which addresses everything from nomenclature for cutlery to automobile tire labeling.?ÿ In this case ANSI partners with another society promulgating standards, the American Society of Mechanical Engineers (ASME).?ÿ See page 8 for different eye-bolt designs.
https://www.rentlgh.com/blog/confused-by-eyebolts-lets-clear-it-up/
