Goat grazing problem

Let tether length R = 160 yds. and silo radius r = R/(2π) yds. The involute in the fourth quadrant is a nearly circular arc. One can imagine a circular segment with the same perimeter (arc length) would enclose nearly the same area; the radius and therefore the area of that segment could be readily computed. The arc length of an involute is given by ${\displaystyle {\tfrac {1}{2}}r\theta ^{2}}$ so the arc length |FG| of the involute in the fourth quadrant is ${\displaystyle {\tfrac {1}{2}}r{\Big [}\theta ^{2}{\Big ]}_{{\tfrac {3\pi }{2}}-\varphi }^{2\pi }}$. Let c be the length of an arc segment of the involute between the y-axis and a vertical line tangent to the silo at θ = 3π/2; it is the arc subtended by Φ. ${\displaystyle c\approx r}$ (while the arc is minutely longer than r, the difference is negligible). So ${\displaystyle |FG|={\tfrac {1}{2}}\cdot 25.46\cdot (6.28^{2}-4.71^{2})+25.46=245.38}$. The arc length of a circular arc is ${\displaystyle r\theta }$ and θ here is π/2 radians of the fourth quadrant, so ${\displaystyle r{\tfrac {\pi }{2}}=245.38}$, r the radius of the circular arc is ${\displaystyle 156.21}$ and the area of the circular segment bounded by it is ${\displaystyle {\tfrac {\pi r^{2}}{4}}=19165}$. The area of the involute excludes half the area of the silo (1018.61) in the fourth quadrant, so its approx area is 18146, and the grazable area including the half circle of radius R, (${\displaystyle {\tfrac {1}{2}}\pi R^{2}}$) totals ${\displaystyle 2\cdot 18146+40212=76505}$. That is 249 sq.yds. greater than the correct area of 76256, an error of just 0.33%. This method of approximating may not be quite so good for angles < 3π/2 of the involute.

If it matters, there is a constructive way to obtain a quick and very accurate estimate of ${\displaystyle \varphi }$: draw a diagonal from point ${\displaystyle {\tfrac {3}{2}}\pi }$ on the circumference of the pond to its intersection on the y-axis. The length of the diagonal is 120yds. because it is ${\displaystyle {\tfrac {3}{4}}}$ of the tether. So the other leg of the triangle, the hypotenuse as drawn, is ${\displaystyle {\sqrt {120^{2}-r^{2}}}=117.27}$ yds. So ${\displaystyle \varphi \approx \sin ^{-1}({\tfrac {r}{117.27}})=.219}$ radians, rounded to three places.

Find the area between a circle and its involute over an angle of 2π to −2π excluding any overlap. In Cartesian coordinates, the equation of the involute is transcendental; doing a line integral there is hardly feasible. A more felicitous approach is to use polar coordinates (z,θ). Because the "sweep" of the area under the involute is bounded by a tangent line (see diagram and derivation below) which is not the boundary (${\displaystyle {\overline {qF}}}$) between overlapping areas, the decomposition of the problem results in four computable areas: a half circle whose radius is the tether length (A₁); the area "swept" by the tether over an angle of 2π (A₂); the portion of area A₂ from θ = 0 to the tangent line segment ${\displaystyle {\overline {tF}}}$ (A₃); and the wedge area qFtq (A₄). So, the desired area A is A₁ + (A₂ − A₃ + A₄) · 2. The area(s) required to be computed are between two quadratic curves, and will necessarily be an integral or difference of integrals.

The primary parameters of the problem are ${\displaystyle R}$, the tether length defined to be 160yds, and ${\displaystyle r}$, the radius of the silo. There is no necessary relationship between ${\displaystyle R}$ and ${\displaystyle r}$, but here ${\displaystyle r={\tfrac {160}{2\pi }}}$ is the radius of the circle whose circumference is ${\displaystyle R}$. If one defines the point of tethering ${\displaystyle v}$ (see diagram, above) as the origin with the circle representing the circumference of the pond below the x-axis, and ${\displaystyle F}$ on the y-axis below the circle representing the point of intersection of the tether when wound clockwise and counterclockwise, let ${\displaystyle t}$ be a point on the circle such that the tangent at ${\displaystyle t}$ intersects ${\displaystyle F}$, and ${\displaystyle |{\overset {\frown }{vt}}|}$ + ${\displaystyle |{\overline {tF}}|}$ Is the length of the tether. Let ${\displaystyle q}$ be the point of intersection of the circumference of the pond on the y-axis (opposite to ${\displaystyle v}$) below the origin. Then let acute ${\displaystyle \angle tFq}$ be ${\displaystyle \varphi }$.

The area under the involute is a function of ${\displaystyle R^{3}}$ because it is an integral over a quadratic curve. The area has a fixed boundary defined by the parameter ${\displaystyle r}$ (i.e. the circumference of the silo). In this case the area is inversely proportional to ${\displaystyle r}$, i.e. the larger ${\displaystyle r}$, the smaller the area of the integral, and the circumference is a linear function of ${\displaystyle r}$ (${\displaystyle 2\pi r}$). So we seek an expression for the area under the involute ${\displaystyle E=f(R^{3}/r)}$.

First, the area A₁ is a half circle of radius ${\displaystyle R}$ so ${\displaystyle A_{1}={\tfrac {1}{2}}\pi \cdot R^{2}.}$

Next, find the angle ${\displaystyle \varphi }$ which will be used in the limits of the integrals below. Let ${\displaystyle x=|{\overline {tF}}|}$. ${\displaystyle \varphi }$ is complementary to the opposite angle of the triangle whose right angle is at point t; and also complementary to that angle in the third quadrant of the circle. ${\displaystyle |{\overline {tF}}|}$ is the unrolled arc ${\displaystyle {\overset {\frown }{vqt}}}$, so its arclength is ${\displaystyle r\theta _{x}}$. So ${\displaystyle \theta _{x}={\tfrac {x}{r}}}$ and ${\displaystyle \theta _{x}={\tfrac {3\pi }{2}}-\varphi }$, so ${\displaystyle \varphi ={\tfrac {3\pi }{2}}-{\tfrac {x}{r}}}$. Finally,${\displaystyle \tan(\varphi )={\tfrac {r}{x}}}$ and the following equation is obtained: ${\displaystyle r=x\cdot \tan {({\tfrac {3}{2}}\pi -{\tfrac {x}{r}})}}$. That is a transcendental equation that can only be solved by trial-and-error, polynomial expansion, or an iterative procedure like Newton–Raphson. ${\displaystyle \varphi \approx 0.21900[+0,-0.00003]}$. [3]

Next compute the area between the circumference of the pond and involute. Compute the area in the tapering "tail" of the involute, i.e. the overlapped area (note, on account of the tangent tF, that this area includes the wedge section, area A₄, which will have to be added back in during the final summation). Recall that the area of a circular sector is ${\displaystyle {\tfrac {1}{2}}r^{2}\theta }$ if the angle is in radians. Imagine an infinitely thin circular sector from ${\displaystyle o}$ to ${\displaystyle p}$ subtended by an infinitely small angle ${\displaystyle \Delta \theta }$. Tangent to ${\displaystyle o}$, there is a corresponding infinitely thin sector of the involute from ${\displaystyle m}$ to ${\displaystyle n}$ subtending the same infinitely small angle ${\displaystyle \Delta \theta }$. The area of this sector is ${\displaystyle {\tfrac {1}{2}}z^{2}\Delta \theta }$ where ${\displaystyle z}$ is the radius at some angle ${\displaystyle \theta _{i}}$, which is ${\displaystyle r\theta _{i}}$, the arc length of the circle so far "unwrapped" at angle ${\displaystyle \theta _{i}}$. The area under the involute is the sum of all the infinitely many infinitely thin sectors ${\displaystyle i}$ through some angle ${\displaystyle \theta _{n}}$. This sum is

${\displaystyle \lim _{n\to \infty }\sum _{i=1}^{n}{\tfrac {1}{2}}(r\theta _{i})^{2}\Delta \theta =\int _{{\frac {3\pi }{2}}-\varphi }^{2\pi }{\frac {1}{2}}(r\theta )^{2}\ d\theta ={\frac {1}{2}}r^{2}\left[{\frac {\theta ^{3}}{3}}\right]_{{\frac {3\pi }{2}}-\varphi }^{2\pi }={\frac {4}{3}}r^{2}\pi ^{3}-{\frac {1}{2}}r^{2}{\frac {({\frac {3\pi }{2}}-\varphi )^{3}}{3}}.}$

The bounds of the integral represent the area under the involute in the fourth quadrant between ${\displaystyle {\overline {tF}}}$ and ${\displaystyle {\overline {vG}}}$. The angle is measured on the circle, not on the involute, so it is less than ${\displaystyle {\tfrac {3\pi }{2}}}$ by some angle designated ${\displaystyle \varphi }$. ${\displaystyle \varphi }$ Is not given, and must be determined indirectly. Unfortunately, there is no way to simplify the latter term representing the lower bound of the eval expression because ${\displaystyle \varphi }$ is not a rational fraction of ${\displaystyle \pi }$, so it may as well be substituted and evaluated at once (factoring out ${\displaystyle \pi }$ preemptively): ${\displaystyle {\tfrac {1}{6}}r^{2}\pi ^{3}\cdot ({\tfrac {27}{8}}-K({\tfrac {\varphi }{\pi }}))}$ which for expository reasons can be rewritten ${\displaystyle A_{2}-A_{3}={\tfrac {R^{3}}{6r}}-{\tfrac {R^{3}}{48r}}\cdot ({\tfrac {27}{8}}-K({\tfrac {\varphi }{\pi }}))}$. It seems apropo to merge a factor of ${\displaystyle {\tfrac {1}{8}}}$ into the constant term to get a common denominartor for the terms, so ${\displaystyle A_{2}-A_{3}={\tfrac {R^{3}}{6r}}-{\tfrac {R^{3}}{6r}}\cdot ({\tfrac {27}{64}}-K({\tfrac {\varphi }{8\pi }}))}$. ${\displaystyle K}$ is dominated by a linear term from the integration, so may be written, ${\displaystyle K={\tfrac {27}{32\pi }}\varphi -Z}$ where ${\displaystyle Z}$ is a non-zero positive but negligible quantity.

A₄ is the area of the peculiar wedge ${\displaystyle \lambda ^{tFqt}}$. That area is the area of a right triangle with vertex t, minus the area of a sector bounded by ${\displaystyle {\overset {\frown }{tq}}}$. ${\displaystyle \lambda ={\tfrac {rx}{2}}-{\tfrac {1}{2}}r^{2}\theta }$ where x is |tF| and θ is the angle opposite to Φ in the right angle triangle. So, ${\displaystyle \lambda ={\tfrac {1}{2}}r(R-r({\tfrac {\pi }{2}}+\varphi ))-{\tfrac {1}{2}}r^{2}({\tfrac {\pi }{2}}-\varphi )={\tfrac {1}{2}}(rR-r^{2}\pi )}$. If ${\displaystyle R=2\pi r}$, then the area ${\displaystyle \lambda }$ of the wedge is ${\displaystyle {\tfrac {1}{2}}\pi r^{2}}$ by reduction.

The final summation A₁ + (A₂ − A₃ + A₄) · 2 is ${\displaystyle {\tfrac {1}{2}}\pi \cdot R^{2}+({\tfrac {R^{3}}{6r}}-{\tfrac {R^{3}}{6r}}\cdot ({\tfrac {27}{64}}-({\tfrac {27}{32\pi }}\varphi -Z))+{\tfrac {1}{2}}(rR-r^{2}\pi ))\cdot 2}$. All imprecision in the calculation is now uncertainty in ${\displaystyle \varphi }$ and the residual ${\displaystyle Z}$. ${\displaystyle \varphi \doteq ({\tfrac {R}{r}}-{\tfrac {\pi }{2}}-\varphi )^{-1}}$. That's useful for elucidating the relationships between the parameters. ${\displaystyle \varphi }$ is transcendental, so the definition is a recurrence relation. The initial guess ${\displaystyle \varphi }$ is a small fraction of ${\displaystyle \pi }$. The numerical answer is ${\displaystyle A=76256}$ rounded up to the nearest square yard.[4] It is worth noting that ${\displaystyle \lim _{\varphi \rightarrow {\frac {\pi }{2}}}A={\tfrac {1}{2}}\pi \cdot R^{2}+{\tfrac {R^{3}}{6r}}}$, which is the answer given for the case where the tether length is half the circumference (or any length such that ${\displaystyle {\tfrac {R}{r}}\leq \pi }$) of the silo, or no overlap to account for. The goat can eat all but 5% of the area of the great circle defined by its tether length, and half the area it cannot eat is within the perimeter of the pond/silo. The only imprecision in the calculation is that no closed-form representation for ${\displaystyle \varphi }$ can be derived from the geometry presented. But small inaccuracies in ${\displaystyle \varphi }$ when ${\displaystyle \varphi \ll 1}$ don't significantly affect the final result.

Just as the area below a line is proportional to the length of the line between boundaries, and the area of a circular sector is a ratio of the arc length (${\displaystyle L=r\theta }$) of the sector (${\displaystyle A={\tfrac {r}{2}}\cdot L}$), the area between an involute and its bounding circle is also proportional to the involute's arc length${\displaystyle L={\tfrac {1}{2}}r\theta ^{2}}$: ${\displaystyle A={\tfrac {r\theta }{3}}\cdot L={\tfrac {r^{2}\theta ^{3}}{6}}}$ for ${\displaystyle 0<\theta <\theta _{i}}$. So the total grazing area is ${\displaystyle A=A_{1}+(A_{2}-A_{3}+A_{4})\cdot 2}$. ${\displaystyle A_{1}={\tfrac {1}{2}}\pi R^{2}}$. ${\displaystyle A_{2}-A_{3}=\left[{\tfrac {r^{2}\theta ^{3}}{6}}\right]_{{\tfrac {3\pi }{2}}-\varphi }^{2\pi }}$.${\displaystyle A_{4}={\tfrac {1}{2}}\pi r^{2}}$.

The area reachable by the animal is in the form of an asymmetric lens, delimited by the two circular arcs.

The area ${\displaystyle A}$ of a lens with two circles of radii ${\displaystyle R,r}$ and distance between centers ${\displaystyle d}$ is

${\displaystyle {\begin{aligned}A={}&r^{2}\arccos \left({\frac {d^{2}+r^{2}-R^{2}}{2dr}}\right)+R^{2}\arccos \left({\frac {d^{2}-r^{2}+R^{2}}{2dR}}\right)\\&{}-{\frac {1}{2}}{\sqrt {(d+r-R)(d-r+R)(-d+r+R)(d+r+R)}},\end{aligned}}}$

which simplifies in case of ${\displaystyle R=d=1}$ and one half of the circle area to

${\displaystyle {\frac {1}{2}}\pi =r^{2}\arccos \left({\frac {1}{2}}r\right)+\arccos \left(1-{\frac {1}{2}}r^{2}\right)-{\frac {1}{2}}r{\sqrt {4-r^{2}}}.}$

The equation can only be solved iteratively and results in ${\displaystyle r=1.1587\ldots }$ (sequence A133731 in the OEIS).

By using ${\displaystyle r<{\sqrt {2}}}$ and integrating over the right half of the lens area with

${\displaystyle {\frac {1}{4}}\pi =\int _{0}^{\sqrt {r^{2}-{\frac {r^{4}}{4}}}}\left({\sqrt {r^{2}-t^{2}}}+{\sqrt {1-t^{2}}}-1\right)\,dt}$

the transcendental equation

${\displaystyle r={\frac {\pi }{\pi r-{\sqrt {4-r^{2}}}+\left({\frac {4}{r}}-2r\right)\arcsin \left({\frac {1}{2}}r\right)}}}$

follows, with the same solution.

In fact, using the identities ${\displaystyle \arccos \left(1-{\frac {r^{2}}{2}}\right)+2\arccos \left({\frac {|r|}{2}}\right)=\pi }$ and ${\displaystyle \arcsin \left({\frac {r}{2}}\right)={\frac {\pi }{2}}-\arccos \left({\frac {r}{2}}\right)}$, the transcendental equation derived from lens area can be obtained.

The area can be written as the sum of sector area plus segment area.[5]

Assuming the leash is tied to the bottom of the pen, define ${\displaystyle \theta }$ as the angle the taut leash makes with upwards when the goat is at the circumference. Define ${\displaystyle \alpha }$ as the angle from downwards to the same location, but from the center of the pen instead of from the center of the larger circle. The sum of angles of a triangle equals ${\displaystyle \pi }$ for the resulting isosceles triangle, giving ${\displaystyle \alpha =\pi -2\theta }$. Setting the pen's radius to be 1 and trigonometry such as ${\displaystyle \sin(2\theta )=2\sin(\theta )\cos(\theta )}$ then give ${\displaystyle \theta =\arccos \left({\frac {r}{2}}\right)}$.

Requiring that half the grazable area be 1/4 of the pen's area gives ${\displaystyle A_{sector}+A_{segment}=\pi /4}$. Using the circular sector and circular segment area formulae gives

${\displaystyle r^{2}\theta +\alpha -\sin(\alpha )={\frac {\pi }{2}}}$,

which only assumes ${\displaystyle 0<r<2}$.

Combining into a single equation gives

${\displaystyle {\frac {\pi }{2}}=r{\sqrt {1-\left({\frac {r}{2}}\right)^{2}}}+(2-r^{2})\arccos \left({\frac {r}{2}}\right)}$.

Note that solving for ${\displaystyle \arccos \left({\frac {r}{2}}\right)}$ then taking the cosine of both sides generates extra solutions even if including the obvious constraint ${\displaystyle 1<r<{\sqrt {2}}}$.

Using trigonometric identities, we see that this is the same transcendental equation that lens area and integration provide.

By using complex analysis methods, Ingo Ullisch obtained a closed-form solution as the cosine of a ratio of two contour integrals:[6]

${\displaystyle r=2\cos \left({\frac {\displaystyle \oint _{C}{\frac {z}{2\sin z-2z\cos z-\pi }}\,dz}{\displaystyle \oint _{C}{\frac {2}{2\sin z-2z\cos z-\pi }}\,dz}}\right),}$

where C is the circle ${\displaystyle \left|z-{\frac {3\pi }{4}}\right|={\frac {\pi }{4}}}$.