Uma Cabra na Corda / Goat on a Rope

Cabro - Rua Prof. Salatiel Torres Bairro Nossa Sra de Lourdes, Ouro Preto

A barn 15m by 25m stands in the middle of a large grassy field. Tied by a rope to one corner of the barn is a hungry goat. Over what area of the field can the goat graze if the rope is:

a. X meters long?

b. 10 meters long?

c. 20 meters long?

d. 30 meters long?

e. 100 meters long?

A farmer owns a circular field of grass (radius=r).

He tethers a goat via a length of rope (R) to the circumference of the field.

What ratio r/R must the farmer choose so that the goat can only eat half the area of grass?

Let O be the center of the circle of the goat's grazing range and let C be the point on the circumference where the goat is tethered.

Let CA and CB be the chords of length R giving the extreme positions on the circumference where the goat can reach.

Angle AOC = angle BOC = phi (radians). We shall first calculate the area of the segment of circular field cut off by the chords CA and CB.

The area cut off is found by subtracting the area of triangle OAC from the sector OAC:

= (1/2)r^2.phi - (1/2)r^2.sin(phi)

= (1/2)r^2[phi - sin(phi)]

The area cut off by both AC and BC is double this: r^2[phi -sin(phi)].

This area must be added to the area of the sector of the circle of radius R between the radii CA and CB.

By simple geometry the angle ACB = (1/2)(2.pi-2phi) = pi-phi

Area of sector CAB = (1/2)R^2(pi-phi)

Total area available for the goat

= r^2[phi-sin(phi)] + (1/2)R^2(pi-phi)

This must equal half the area of the field (1/2)pi.r^2

r^2[phi-sin(phi)] + (1/2)R^2(pi-phi) = (1/2)pi.r^2

Dividing through by (1/2)r^2: 2[phi-sin(phi)] + (R/r)^2(pi-phi) = pi

Now we can get another relationship between r and R by drawing a perpendicular from O to AC to bisect AC.

This shows that: r.sin(phi/2) = R/2 and R/r = 2sin(phi/2)

Equation(1) can be written: 2[phi-sin(phi)] + 4sin^2(phi/2)[pi-phi] - pi = 0

This can be solved for phi by Newton Raphson or by a handy TI-92 calculator to give: phi = 1.23589 radians

Then R/r = 2sin(.617945) = 1.158723

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Bibliografia

Kasimatis, E., Kysh, J., Salle,T., & Hoey, B. Undertanding Mathematics: Data, Equations, and Graphs - Making Sense of Elementary Algebra. (ISBN 0-201-76800-3). Addison-Wesley: New York. 1997.