Calculating the Grazing Area of a Goat: IB Mathematics SL Project

This project addresses the classic goat grazing problem, where a goat is tethered to a silo and barn corner. The solution involves calculating the grazing area using calculus, specifically Green's Theorem. The student begins by illustrating the goat's motion and the involute trajectory formed as the chain wraps around the silo. The area calculation involves establishing parametric equations for the involute and applying Green's Theorem to determine the grazing area. The project provides detailed diagrams, calculations, and references, demonstrating a clear understanding of the mathematical concepts and a methodical approach to solving the problem. The final grazing area is calculated, and the solution showcases the application of calculus in a practical context.

Q.
Given: A retired mathematics professor has decided to raise a goat.He
owns a silo and a barn.The barn’s front wallis tangent to the silo at the
corner.The silo has a circular base with a radius of 10 feet.The professor
has decided to tether the goat to a chain that is anchored at the corner of
the barn, the point of tangency.He has also cut the chain so that it is long
enough to wrap around the silo exactly once, that is, the length of the chain
equals the circumference of the silo.The barn length is longer than the chain.
base radius ofsilo = 10 feet,length ofchain = circumference ofsilo =
2π r = 20π feet.
Approach to Solution:
For the goat to maximize it’s grazing area,it starts with the chain out
stretched along the front wallof the barn and then moves in a circular di-
rection towards the filed.As the goat moves in this way, its chain, tethered
to the corner, wounds around the silo.This is demonstrated in figure-1 below.
The goat,with this motion,moves in the grazing field as its chain is be-
ing continuously wound around the silo.It comes to a stop when allthe
chain length is exhausted, which is when it completely wrapped around the
silo. The goat at this point is located at the same corner from which it
started.Figure-2 shows this complete motion of the goat.This type of mo-
tion is called involute of a circle (silo is the circle here), and it is obtained by
winding (or unwinding) a string around a circle.
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Figure 1:Goat’s motion in the field
The invloute trajectory of the goat is drawn below (figure-2) by considering
the chain is tightly wound around the silo and then slowly unwound (as the
goat moves) until it is straightly aligned along the front wall of the barn.
Figure 2:Goat’s trajectory - Involute
Grazing area of the goat:
The grazing area ofthe goat willthe the area inside the trajectory ofthe
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goat (involute) minus the area ofcircle (silo),as seen in figure-2.To cal-
culate this area we willmake use of the Green’s Theorem (Calculus).But
before that, the trajectory (involute) needs to be expressed as a parametric
equation.This part will be clear once we start computing the area of grazing.
Parametric equation of invloute :
Refer figure-3 below.A string,suppose,is wound around a circle with ra-
dius a, and its free end is point P (a, 0).The string is unwound (clockwise),
stretched so the portion RS (unwound string) is tangent at R.
Figure 3:Parametric form of involute
The radius makes an angle x with the horizontal.Since the motion is clock-
wise,using t as a parameter,we equate thisangle to 2π − t. The co-
ordinates ofpoint R are (a cos x, −a sin x),which in parametric form will
be (a cos (2π − t), −a sin (2π − t)).
RS is the length of unwound string,which is same as the length of arc un-
wound on the silo, therefore RS = ax = a(2π − t).
The x- coordinate of S therefore will be:
a cos (2π − t) + RS sin x = a cos (2π − t) + a(2π − t) sin (2π − t)
Similarly, the y- coordinate is:
a sin (2π − t) + RS cos x = −a sin (2π − t) + a(2π − t) cos (2π − t)
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Noting that,sin (2π − t) = − sin t,and cos (2π − t) = cos t,finally we get
the simplified form as:
r(t) = ah(cos t − (2π − t) sin t,sin t + (2π − t) cos ti
Area of grazing by Calculus :
Green’s Theorem [1]:Area bounded by a closed curve C is evaluated by
computing the line integral:
1
2
Z
C
−y dx + x dy
where, C is oriented counter-clockwise.
The area of grazing covered by the goat is bounded by curves as shown in
figure below:
Figure 4:Curves and area of integration
C1 is the front wallof the barn,C2 is the trajectory of the goat (involute)
and C3 is the circular curve (silo).
Parametric curves for each of the curves is given below:
C1 : r(t) = ah1,ti
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C2 : r(t) = ah(cos t − (2π − t) sin t,sin t + (2π − t) cos ti
C3 : r(t) = ahcos (2π − t),sin (2π − t)i
C1 and C3 are standard curves (line and circle), so their parametric equations
are fairly straight forward.
The line integral for area of curve is calculated in the following way:
1
2
Z
C
−y dx+x dy =
1
2
Z
C1
−y dx+x dy +
1
2
Z
C2
−y dx+x dy +
1
2
Z
C3
−y dx+x dy
Also, 1
2
Z
C
−y dx + x dy =
1
2
Z
C
h−y,xi · dr
Also, 0 ≤ t ≤ 2π
Now,
1
2
Z
C1
−y dx + x dy =
1
2
Z
C1
h−y,xi · dr
[r(t) = ah1,ti =⇒ x = a & y = at,dr = ah0, 1i dt]
= 1
2
Z 2π
0
h−at,ai · ah0, 1i dt
= a2
2
Z 2π
0
dt
= a2
2
h
t
i 2π
0
= π a2
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Similarly,
1
2
Z
C3
−y dx + x dy =
1
2
Z
C3
h−y,xi · dr
= a2
2
Z 2π
0
− sin2 (2π − t) − cos2 (2π − t) dt
= −a2
2
Z 2π
0
dt = −π a2
And,
1
2
Z
C2
−y dx + x dy =
1
2
Z
C2
h−y,xi · dr
= a2
2
Z 2π
0
[sin t + (2π − t) cos t] (2π − t) cos t
− [cos t − (2π − t) sin t] (2π − t) sin t dt
= a2
2
Z 2π
0
(2π − t)2 dt
= 4
3π3 a2
Area of grazing, by Green’s Theorem is therefore given by:
Area =1
2
Z
C
−y dx + x dy = π a2 − π a2 + 4
3π3 a2 = 4
3π3 a2
Since the radius of silo is 10 feet, substituting that, we get:
Area =4
3π3 102 = 400
3 π3 feet2
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References:
[1] J. Orloff, ”Line Integrals and Green’s Theorem,” MIT-OCW, May, 2011.
[Online].Available:MIT-OCW, https://math.mit.edu/ jorloff/18.04/notes/greenstheorem
[Accessed June 2, 2019].
[2]E. Weisstein,”Goat Problem.” From Math World-A Wolfram Web Re-
source, 2016.[Online].Available:http://mathworld.wolfram.com/GoatProblem.html.
[Accessed June 2, 2019].
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