Final Exam part 1: Goat Grazing Area
by
Molade Osibodu
Question: 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.
Solution:
The figure below shows the field with the shed and silo in place
The figure below however shows a better picture of the grazing area. To see a much nicer picture, click here to see the work of Megan Dickerson
PART A
The picture above shows two sectors that are a fourth of the circle. The length is 76.7-10 = 66.7 ft. Therefore, the area is
To find the area of the triangle, we need to calculate the height. Since it is a right-angled triangle, it is
So the area of the triangle is
If you look at Megan's page, there are two sectors left in part A which my picture didn't do much justice. To find the angle formend by the sector, we can use the law of cosine to find the angle of the triangle first. The angle is
So the angle formed by the sector is 90 - 77.63 = 12.37
Therefore, the area of the two identical sectors is
Hence, the total area formed in part A is 6988.3 + 456 + 470.8 = 7915.1 sq ft
PART B
This part is very easy to calculate. It is simply 153.4 * 92 = 14112.8 sq ft
PART C
This part was approximated by calculating the area of the two sectors and subtracting the area of the silo
Thus, the total grazing area for the goat is 6674.14 + 7915.1 + 14112.8 = 28702.04 sq ft