Prof Tom Dreyer from the Dept. of Applied Mathematics, University of Stellenbosch, used the following delightful problem in the late 70's and 80's to give several talks to teachers about 'mathematical modelling':
Jannie's father writes to his son, Jannie, who is in a hostel in town to attend high school, asking him to determine how one of their sheep grazing camps can be divided into two camps of equal area. The dividing fence needs to start from the shared windmill on the western side of the camp (so as the provide water for both camps) - see map below.
a) Assuming as a first rudimentary model that the terrain is completely flat, one can easily model the problem dynamically as shown in the sketch below by using dynamic geometry and importing a copy of the map, and then making appropriate constructions and measurements.
b) Drag F to find a point where the area of the camp is halved, and determine using the given map scale, the amount of fencing required. Do you think this value for the length of fencing needed is exact, an under-estimate or an over-estimate? (Explain your reasoning).
Jannie's Father's Farm Problem
1) Clearly the above solution for the required length of fencing, EF, is an under-estimate since it doesn't allow for the elevation up and down the hill (Korhaankop). Adapt your answer for EF by using the given contours in the map to determine the elevation which goes from about 155 m at the windmill to about 263 m at the heighest point of the hill, and then down again to about 175 m at F. This should help you more accurately determine the length of fencing required.
2) However, taking the fencing over the hill might not be a good choice as putting in poles in a hard rocky hill might be difficult. Explore and suggest other possible alternative solutions.
Read Prof Tom Dreyer's eventually published paper (in Afrikaans) at Jannie se pa se plaasprobleem.
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Created by Michael de Villiers, 16 June 2016.