grazing goat problem, how much of the area can the goat graze ?

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pauli relanderi
pauli relanderi on 16 Oct 2021
Edited: DGM on 16 Oct 2021
Hello !
Im just wondering does anyone know how would i go about solving this in matlab
How much of the fields area can it reach to eat ?
when
r is 0.8 %goats leash
R is 1 %the big circler radius
ive tried calculations i found online and got nowhere
This is what ive gotten
clear all
clc
r=0.8
R=1
d=R
alfa=acos(r./(2.*R))
A=(r.^2-(2.*R.^2)).*alfa+R.^2.*pi-R.^2.*sin(2*alfa)
Buts this gets me answer of 0.8318 which is too much it should be around 2-3, atleast i think it should
  1 Comment
Star Strider
Star Strider on 16 Oct 2021
What is the problem statement, and the goat’s constraints (other than the leash length)?

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Accepted Answer

DGM
DGM on 16 Oct 2021
Edited: DGM on 16 Oct 2021
Your answer is reasonable. As a point of verification:
rf = 1;
rg = 0.8;
% intersection point locations
% could also use circcirc()
xint = (2*rf^2 - rg^2)/(2*rf);
yint = sqrt(rf^2 - xint^2);
% sector areas
secareaf = rf^2 * asin(yint/rf);
secareag = rg^2 * asin(yint/rg);
% sum of triangle areas
triarea = yint*(xint + sqrt(rg^2 - rf^2 + xint^2));
% total area
totalarea = secareaf + secareag - triarea
totalarea = 0.8318
Furthermore, it's helpful to check things visually. Consider the simplified case where rf = 1 and rg = 1.
Given that the area of either circle = pi, then visually we know Aintersection < pi/2 ~ 1.5708
and considering the inscribed triangles, we know Aintersection > sqrt(3)/2 ~ 0.8660
Running the above code for this second case gives a result of 1.2284, which is indeed within those boundaries. Looking back at the original problem with the smaller rg, we should expect Aintersection to be correspondingly smaller than 1.2, even if the geometry becomes less obvious.

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