1.0 AIM
3
1.1 SCOPE
3
2.0 METHOD
3
3.0 DEVISING THE MODEL
4
4.0 COMMENTS
6
5.0 CONCLUSION
6
6.0 References
7
Appendix 1, Figure 1: sketch of paddocks showing both fencing options
8
Appendix 2, Figure 2: rows of width, length and area from spread sheet
9
Appendix 3, Figure 3: graphs of area vs. width for fencing option 1 and 2
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1.0 AIM
The objective is to use mathematical techniques to determine maximum area of land that can be fenced using a given size of fencing material. An equation will be obtained to describe the area that gives maximum benefit.
1.1 SCOPE
The scope of the work is to find out the maximum area that can be fenced using wire and posts that can only cover a perimeter of 5000 meters. The area of paddock to be fenced is rectangular and two options are available. The paddock three sides of the rectangular paddock could be fenced with the fourth side being a gulley. The yield from this part of land would be $11000/ha.
Second option would be fencing four sides on an area that would yield $18000/ha. These two options will be shown graphically and the mathematical equation will establish the best option.
2.0 METHOD
1. For each of the two locations on the farm, draw a neat sketch of the paddock, labeling length L and width W.
2. Create a spreadsheet which shows widths going up by a certain amount, and for each width, show resulting length and area of rectangular paddock.
3. For each paddock, you will need to come up with the appropriate formulae for length and area, bearing in mind that area must be in hectares.
4. For each paddock, supply one printout without formula and one with formula.
5. Plot the two paddock areas against the width in a chart generated from the spreadsheet. The two charts may be created separately or together on one chart. Use the spreadsheet program to find the equation for each of the curves of best fit.
6. Identify the width, length and area that yield maximum area for each of the two paddocks.
7. Calculate the yield that could be generated from the two maximum areas identified in the previous step.
8. Write up your findings in a report to meet the Assessment Criteria, with relevant explanations and justifications to ensure it can be understood by anyone with suitable mathematical background.
3.0 DEVISING THE MODEL
WHY CHOSEN
All the methods outlined above were followed. It was important because two results were to be compared. A sketch of the paddocks is drawn in figure 1 in appendix 1.
FINDING THE MODEL
1.0 The equations for area and perimeter for both fencing options was obtained as below and the length was expressed in terms of width. The area was also expressed in terms of one variable: width
Option 1 (gulley):
Option 2 (full fence):
2.0 From the above relations of area and perimeter, a schedule was generated in excel for dimensions and area. The area was converted from square meters into hectares using the below relation.
3.0 The width in excel is to go in steps of 50m. This is reasonable to get fairly many points to generate a smooth curve, but not too many to occupy a large space. A total of 25 points was obtained.
4.0 The range of width used in excel was from 50 m through 1500m. 0m could not be used because it is impractical. 50m was chosen for a start because it is reasonable width dimension given that the perimeter is 5000m. Practically, the maximum width occurs when W=L. in this case, the maximum W=1250m. If W>L, then it is no longer the width, it is the length. However, for purpose of monitoring the effect, width is allowed to ‘exceed length’ so as to see on the effect on the curve.
5.0 the rows from the spread sheet are shown in figure 2 of appendix 2.
6.0 from the graph of area vs. width, the maximum area that can be obtained is . This is possible under option 1 where one side of the farm is a gulley. The dimensions are:
7.0 the equation of the curve that yields maximum area is given as:
8.0 from the spreadsheet, maximum area under the first option with a gulley on one side yielded the most. The amount was calculated as for each option.
The maximum yield was $3,437,500.
4.0 COMMENTS
The method of using a spread sheet is convenient in finding the relation between two variables. As is the case in the above, fencing problem, spreadsheet was handy in realizing the largest area and the maximum yield.
A possible error could arise in reading the maximum point of the graph but this was taken care of in this case because a value in the table coincided with the maximum value on the graph.
5.0 CONCLUSION
The fencing problem was amicably solved. It was realized that with one side not requiring a fence, the area that could be fenced by 5000m length of fencing material was more and produced more yields as well. Using spread sheet is, therefore, a versatile method of solving area problems that are constrained by a given perimeter.
6.0 References
Project Criterion guide; Use mathematics in a range of contexts
Appendix 1, Figure 1: sketch of paddocks showing both fencing options
Appendix 2, Figure 2: rows of width, length and area from spread sheet
option 1 (gulley)
yield ($)
option 2 (full fence)
2W+L=5000
option 1
option 2
A (ha)
gulley
full fence
W
L
A (ha)
Appendix 3, Figure 3: graphs of area vs. width for fencing option 1 and 2
