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 ft^{2} |
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 ft^{2}
2 (3494.2 ft^{2}) = 6988.3 ft^{2}
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 ft^{2}
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 r^{2}θ = 1/2 (46.7
ft)^{2} (0.23 radians) = 250.8 ft^{2}
2 (250.8 ft^{2}) = 501.6 ft^{2}
The
total potential grazing area around the shed is the sum of these areas:
Area (shed) = 6988.3 ft^{2} + 456.2
ft^{2} + 501.6 ft^{2} = 7946.1 ft^{2}
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 = Area_{center} + Area_{shed}
+ Area_{silo}
Area = 14112.8 ft^{2} + 7946.1 ft^{2}
+ 7603.7 ft^{2} = 29662.6 ft^{2}
Home |
EMAT 6680
| Dr. Jim
Wilson