I Believe this to be a definitive solution to the infamous problem of a goat tied to the perimeter fence of a circular field. The problem requires the solver to determine the length of goat tether that enables the goat to eat a maximum of one half of the area of grass within the field. Generally, the area of the field is given; however, it is the radius of the field that matters and this may be calculated from radius = √(area/π). Furthermore, since all circles are similar the solution can be based on a field of unity radius and the length of the tether determined as a ratio of the field radius.
Referring to the diagram below, the point O denotes the centre of the field and the point P depicts the point on the perimeter where the goat is tethered. When the goat tether is fully extended it forms an arc between points A and B and together with the perimeter fence marks out the area able to be grazed (coloured yellow). The perpendicular, from A joins horizontal field radius OP at pointC, used later to determine the area of kite shape OAPB (being 2 x triangle OAP). The distancesAP and BP represent the radial r length of the goat tether. Angle POA is equal to angle POB and is denoted by θ; the principal variable for my solution put forward here.
The portion of the field not grazed (coloured green) is equal to that which is grazed (coloured yellow) by the goat. In a field of unity radius both portions have an area of π/2 square units. Field segment OAB includes all of the ungrazed area and some of the grazed area. To ascertain just the ungrazed area, it is necessary to remove the area formed by the goat tether segment APB and then give back the area of the kite shape OAPB. The length of AC is clearly sinθ, length OC similarlyis cosθ, and area triangle AOP is therefore 1/2sinθ.
Area of any segment of a circle is 1/2δR2; where δ is the radial angle of the segment and R is the radius.
The larger field segment bounded by OAB has an area (2π-2θ)/2 = π-θ
Field segment bounded by PAB has area r2(2π-2θ)/2*2 = r2(π-θ)/2
Twice area of triangle OAP; that is, 2sinθ/2 = sinθ
Length of CP = 1-cosθ; then by Pythagoras:
r2 = sin2θ+(1-cosθ)2
= sin2θ+(1+cos2θ-2cosθ) and since sin2+cos2 = 1
π/2 = (π-θ)-1/2(π-θ)r2+sinθ = (π-θ)-(π-θ)(2-2cosθ)/2+sinθ
= (π-θ)-(π-θ)(1-cosθ)+sinθ = (π-θ)-(π-θ)+(π-θ)cosθ+sinθ
π/2-sinθ = (π-θ)cosθ becomes (π-2sinθ) = 2(π-θ)cosθ
and then cosθ = (π-2sinθ)/2(π-θ) Finally: θ = arccos((π-2sinθ)/2(π-θ))
The above is now solved iteratively using θk = arccos((π-2sinθk-1)/2(π-θk-1))
Since θ, in this case, must lie between π/2 and π/3 radians a good initial guess for θk should be 5/12πradians. Using a good calculator the iterations are:
θ = 5/12π
θ = arccos((π−2sin 1.3089969389957472)/2(π−1.3089969389957472)
θ = arccos((π−2sin 1.2344269017829694)/2(π−1.2344269017829694)
θ = arccos((π−2sin 1.2358963578379363)/2(π−1.2358963578379363)
θ = arccos((π−2sin 1.2358969242798252)/2(π−1.2358969242798252)
θ = arccos((π−2sin 1.2358969242799094)/2(π−1.2358969242799094)
The above iteration has very quickly zoomed in on an extremely accurate answer. The result is now used to determine the goat tether length in relation to the unity radius of the field; as follows:
r = √(2-2cos 1.23589692428)