Grazing Area

Rozina Essani

Farmer Jones had a goat on
a tether. He tied the end of the tether not attached to the goat to a stake in
a field. Over what area could the goat graze? Of course you need to know
something about the length of the tether and about the field.

There are two structures in
the field:

-- a shed that is 20 ft long and 20 feet wide (square)

-- a silo that is 20 ft in diameter

The center of the shed and
the center of the silo are on a line and the distance apart is 92 feet. The
distance from center to center, if you wanted to use this data, is 112 feet.

The tether for the goat is
76.7 feet long. The stake to which the tether is tied is somewhere along the
line of centers between the shed and the silo.

Explore the area over which
the goat can graze as the stake is moved along this line segment from the
midpoint of the side of the shed to the edge of the silo.

__Area around the shed__

We can get the area
around the shed in three parts, A1 = (area of the ¼ circle with radius
66.7ft) x 2, A2 = (area of green arc) x 2, and A3 = (area of blue triangle) x
2.

A1 = 2 x ¼ area
of circle

= 1/2 pr^{2}

= ½ p (66.7)^{2}

= 6988.3ft^{2}

A3:

h^{2} = 46.7^{2} - 10^{2}

h = 45.6ft

A3 = (½
(45.6)(10))*2

= 456 ft^{2}

A2:

We can find the arc
angle by find the angle of the blue triangle opposite h and then subtracting it
from 90 deg.

cos q = 10/46.7

cos q = 10/46.7

q = 77.63¡

arc angle = 90 – 77.63

= 12.37¡

A2 = ((12.37/360) * p * r^{2}) * 2

= (0.034(p)(46.7^{2})) * 2

= 465.9 ft^{2}

Total grazing area
around shed = A_{shed} = A1 + A2 + A3 = 6988.3 + 465.9 + 456 = 7910.2
ft^{2}

^{ }

__Rectangular Area between shed
and silo:__

A = 2 * (92)(76.7)

= 14,112.80 ft^{2}

__Area around silo:__

To find the grazing
area around the silo, first let us break the circle into p/4¡ sectors. By doing this we can find the arc length of
each sector and hence know how much to deduct from the original length of the
tether. By finding this we can then approximate the area of triangles. We know
that the radius of the circle is 10 ft.

Arc length = rq

= 10(p/4)

So each arc length is
7.85 ft.

Arc Angle |
Arc Length = rq |
Tether Length –
Arc Length |

p/4 |
10(p/4)=7.85 |
76.7-7.85=68.5 |

p/2 |
10(p/2)=15.7 |
76.7-15.7=61 |

3p/4 |
10(3p/4)=23.56 |
76.7-23.56=53.14 |

p |
10(p)=31.42 |
76.7-31.42=45.28 |

Using this data let us
now estimate the areas of the triangles. We will have two cases. In the first
case we will start with length 76.7 and in case two we will start with length
68.5.

Case 1:

Area = (q/2)r^{2}

Area_{1} = p/8(76.7)^{2} + p/8(68.5)^{2} + p/8(61)^{2 }+ p/8(53.14)^{2}

= 6723.01 ft^{2}

Area_{2} = p/8(68.5)^{2} + p/8(61)^{2 }+^{ }p/8(53.14)^{2} + p/8(45.28)^{2}

=
5217.95 ft^{2}

average Area_{silo} = [(6723.01+5217.95)/2] x 2

= 11941 ft^{2}

**Total Grazing Area = A _{shed}
+ A + Area_{silo}**

** =
7910.2 + 14112.8 + 11941**

**
= 33,964 ft ^{2}**