Goat in a Field
Given a square building with each side 20 ft. in a field. A goat is tethered to a point on the base of the building. If the tether is 40 ft. long, what is the area over which the goat can graze?
What if the tether is 50 ft. long?
Does it matter where the tether is tied to the side of the barn? That is, will the area be the same regardless where the tether is connected? Why?
Here is one model:
Discussion: Setting up some algebra to represent the semicircular region and the four quarter circle regions should make it possible to have a function for the area. Since this is about area, one might expect a quadratic function over some particular interval. Let x = the distance from one corner of the shed, say the lower left corner in the model above. Then the five radii are 40, 40 - x, x, 40 - x - 20, and 40 - (20 - x). [If that is hard to see let the two pieces on the side of the shed be x and y and we have y = 20 - x]. Now write the expression and simplify.
A graph of the function using a graphing utility will show the quadratic nature of the function. The symmetry of the situation would argue that the midpoint of the interval (i.e. x = 10) is either the minimum value or the maximum value and the two endpoints will be the same, either both the minimal value or both the maximal value.
Generate a GSP file to model the situation.
EXTENSION:
What if the goat was tethered to the base of a SILO?
To simplify the problem, assume the tether is one-have the circumference of the base of the silo. A typical silo such as pictured would have a radius of 6 to 10 feet. Assume the radius is 10 feet so one have the circumference is 10π.
Silo's are farm storage strutures that still dot the rural landscape but are seldom used in modern agriculture. Many would not have the metal dome on the top.