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 pr²

= ½ p (66.7)²

= 6988.3ft²

A3:

h² = 46.7² - 10²

h = 45.6ft

A3 = (½ (45.6)(10))*2

= 456 ft²

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 * r²) * 2

= (0.034(p)(46.7²)) * 2

= 465.9 ft²

Total grazing area around shed = A_shed = A1 + A2 + A3 = 6988.3 + 465.9 + 456 = 7910.2 ft²

Rectangular Area between shed and silo:

A = 2 * (92)(76.7)

= 14,112.80 ft²

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)r²

Area₁ = p/8(76.7)² + p/8(68.5)² + p/8(61)² + p/8(53.14)²

= 6723.01 ft²

Area₂ = p/8(68.5)² + p/8(61)² + p/8(53.14)² + p/8(45.28)²

= 5217.95 ft²

average Area_silo = [(6723.01+5217.95)/2] x 2

= 11941 ft²

Total Grazing Area = A_shed + A + Area_silo

= 7910.2 + 14112.8 + 11941

= 33,964 ft²