Feed that Goat of yours!
By Stephanie Britt
I have been given a very difficult problem of figuring out the largest grazing area for a goat that is tethered to rope between a barn and a silo. Given the length of the tether and the distance between the buildings where is the best place to stake the tether to maximize the grazing area?
The area of the silo is less than the area of the barn so to maximize the feeding grounds we must eliminate the barn from the feeding area. In order to do this the tether must be the full length away from the barn, 76.7 feet.
The total area possible for a goat to graze with a tether length of 76.7 feet is 18481.64 sq feet. If we subtract the area of the silo (314.16 sq ft) and area that is shortened by the silo (136.59 sq ft) the most the goat can graze is 18030.89 square feet.
A Picture is shown below to show how this number was calculated.
If we consider vector from the center of the circle to the outer ring and subtract the tether shortening while wrapping around the silo then we can figure out the maximum area of grazing. Realistically the tether creates an isosceles triangle from the center. The tether will shorten as it wraps around the silo, but if we consider the largest amount that can be taken away is the white area behind the silo then we can see the "minimumized" maximized amount of grazing area. If we find the area of the "pie piece" and find th area of the triangle we can find the area that the goat can not cover and calculate the total grazing area.
Area of Large Circle = 18481.64
Area of "Pie Piece"= 2392.35
Area of Triangle = 2137.05
Area that it can not graze behind silo=255.3
Area of Silo=314.16
So the total area that it can graze is 18481.64-(255.3+314.16)=17912.18 sq ft
If you would like to explore the area yourself or create a new feeding area the GSP is below.