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The goat grazing problem

Anonymous Mon 28 Feb 05:20:36 2022 No.14269209 View Reply Original Report

Quoted By: >>14269215 >>14269240 >>14269252 >>14269264 >>14269519 >>14269527

How do you solve this?

>goat is tied to the side of a circular barn with a rope
>the radius of the barn is 10 meters and the length of the rope is exactly 10 pi meters
>what is the exact surface area on which the goat is allowed to graze?

Anonymous

Anonymous Mon 28 Feb 2022 05:22:33 No.14269215 Report

Quoted By: >>14269218 >>14269221

>>14269209
is the base of the rope fixed in place or does it rotate with the grope?

Anonymous

Anonymous Mon 28 Feb 2022 05:24:21 No.14269218 Report

Quoted By:

>>14269215
probably fixed in place, its too easy otherwise

Anonymous

Anonymous Mon 28 Feb 2022 05:25:25 No.14269221 Report

Quoted By:

>>14269215
It is fixed in place. I chose 10 pi meters for the rope because then the goat can just barely make it to the opposite side of the barn and it makes the problem interesting.

Anonymous

Anonymous Mon 28 Feb 2022 05:34:45 No.14269234 Report

Quoted By: >>14270367

Well, you've basically defined the goat's available area as the space between two involutes of the circle with nice, convenient parameters, so look up / derive the parametric equations for the involute of a circle and integrate between them, then subtract the area of the circle.

Anonymous

Anonymous Mon 28 Feb 2022 05:38:59 No.14269240 Report

Quoted By: >>14269253 >>14269634

>>14269209
I'm too lazy to solve rn, but I'm pretty sure this can be solved via integration? The max distance from the circle the goat can graze decreases to 0 and I'm sure the exact function governing the difference in distance between the edge of the circle and the max distance the goat can move out as a function of theta can be determined. Then it's just a matter of integrating this from 0 to pi and then doubling it?

Anonymous

Anonymous Mon 28 Feb 2022 05:45:45 No.14269252 Report

Quoted By:

>>14269209
https://en.wikipedia.org/wiki/Goat_problem

Anonymous

Anonymous Mon 28 Feb 2022 05:46:31 No.14269253 Report

Quoted By:

>>14269240
this.

Anonymous

Anonymous Mon 28 Feb 2022 05:54:35 No.14269264 Report

Quoted By: >>14269300

>>14269209
Why isnt it just 100pi^2 - 100pi
Just the amount of reach land in total minus the barns size

Anonymous

Anonymous Mon 28 Feb 2022 06:20:02 No.14269300 Report

Quoted By:

>>14269264
because the rope gets wrapped around the barn

Anonymous

Anonymous Mon 28 Feb 2022 09:59:21 No.14269519 Report

Quoted By:

>>14269209
Barns aren't round
By a lawnmower
maths isn't real

Anonymous

Anonymous Mon 28 Feb 2022 10:10:22 No.14269527 Report

Quoted By:

>>14269209
The rope will 1/2 circumference shorter when you go to the furthest point from staring.

The area will be integrated between the two extremes. Final area will be like an egg with a hole in it

Anonymous

Anonymous Mon 28 Feb 2022 12:14:37 No.14269634 Report

Quoted By:

>>14269240
Back to solve I guess, if we let the barn radius be R, and the length of the rope be L, I'm pretty sure we can assume that the rope perfectly hugs the circumference of the barn as the angle from 0 increases. The max distance from the barn at any angle should be the length of the rope minus arclength of the barn from 0 to whatever angle or L - R?. Integrate this from 0 to pi and double this, you get an area of about 99 m^2

Anonymous

Anonymous Mon 28 Feb 2022 19:11:49 No.14270367 Report

Quoted By: >>14270446

>>14269234
What is an involute?

Anonymous

Anonymous Mon 28 Feb 2022 19:36:14 No.14270446 Report

Quoted By:

>>14270367

7:01
https://youtu.be/Q-XOM4E4RZQ?t=419

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