Final Project :: Grazing Area

The Cycloid

By Jamie K. York

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? Note that there are two structures in the field: a shed 20ft by 20ft and a silo with a diameter of 20ft. The center of the shed and the center of the silo are on a line, 92 ft apart. The tether for the goat is 76.7 ft long. The stake to which the tether is tied is somewhere along the line of centers between the shed and the silo.

 As the tether stake moves along the line between the shed and the silo, there is a large area than could potentially be accessed by the goat. We are given that the furthest two points where the stake can be placed are at the shed or at the silo. The length of the tether, the radius, and the stake, circle center, form a circle. As these circles change with various stake placements, we can find a rectangular figure in the center. This figure is made up of the distance between the shed and silo, 92 ft, and the diameter of the circle formed by the tether, 153.4 ft. This rectangle has an area as noted to the right. (92 ft)(153.4 ft) = 14112.8 ft2

Exploring the area near the shed, we find that the tether length is shorten as the goat wonders around the shed. We can use areas of circles to aid in our area calculations. The first area is directly to the right and left of the shed. With a she length of 20 ft, and the tether centered on its edge, the roaming length for the goat is cut short my 10 ft. The area of each of these sides is a quarter of the size of the circle formed by the shortened length, 66.7 ft. Thus, the area of each of these is

1/4 ¹ (66.7 ft)2 = 3494.2 ft2

2 (3494.2 ft2) = 6988.3 ft2

There is still one area that remains on the far side of the shed. This area can also be explored using areas of circles. These circles are centered at the far corners of the shed. In calculating the area, however, we must not double count these two circles, nor the area already covered by the green portion. These orange circles are of radius 46.7 ft, as 30 ft has been used in the goat getting around the shed. The area of the center, isosceles triangle formed calculated using the measures we know, the radius of the orange circle and the side of the shed. We can use Heron's Formula to calculate the area from here.

Area =

s = (46.7 ft + 46.7 ft + 20.0 ft) / 2 = 56.7 ft

Area =  = 456.2 ft2

To find the area of the remaining sectors, we can use the sector area formula. On GSP, we can calculate the angle measure, which is 12.97 degrees or 0.23 radians.

Area = 1/2 r2θ = 1/2 (46.7 ft)2 (0.23 radians) = 250.8 ft2

2 (250.8 ft2) = 501.6 ft2

The total potential grazing area around the shed is the sum of these areas:

Area (shed) = 6988.3 ft2 + 456.2 ft2 + 501.6 ft2 = 7946.1 ft2

The area around the silo requires a slightly different approach since the silo has a circular base. As the goat stretches to the maximum length walking around the silo, he will slowly shorten the tether as part of it begins to wrap around the silo. The portion by which the tether is shortened is equal the arc length that the tether is covering. The remainder of the tether, or the distance from the silo, is stretched tangent to the silo wall. This image is mirrored on the other side of the silo.

We can estimate the area formed by these two arcs and the silo by using triangles. We can average our underestimate, using the four smaller lengths, and overestimate, using the four larger lengths. This is then doubled to estimate the area over the full half circle.

 Angle Silo Length (s) s = rθ, r = 20 ft Tether Length (t) t = L - s, L = 76.7 ft 0 (20 ft)(0) = 0 ft 76.7 ft - 0 ft = 76.7 ft ¹/4 (20 ft)(¹/4) = 15.7 ft 76.7 ft - 15.7 ft = 61.0 ft ¹/2 (20 ft)(¹/2) = 31.4 ft 76.7 ft - 31.4 ft = 45.3 ft 3¹/4 (20 ft)(3¹/4) = 47.2 ft 76.7 ft - 47.2 ft = 29.5 ft ¹ (20 ft)(¹) = 62.8 ft 76.7 ft - 62.8 ft = 13.9 ft

Underestimate =

Overestimate =

Average =

The total estimated area that the goat could potentially cover, with the given limitations on where the tether stake can be placed is sumation of the areas which we have calculated.

Area = Areacenter + Areashed + Areasilo

Area = 14112.8 ft2 + 7946.1 ft2 + 7603.7 ft2 = 29662.6 ft2