Mathematics SL Portfolio Type II. This portfolio considers commercial fishing in a country in two different environments, namely the sea and fish farms. The statistics are obtained from UN Statistics Division Common Database from the year 1980 to 2006. T
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Introduction
Contents
Introduction
Part I
Defining Variables, Parameters and Constraints
Plotting Graph
Trends in Graph and Model Suggestion
Part II
Analytical Development of a Suitable Model for to Fit the Data Points
Part III
Model Function and Original Data Points
Part IV
Plotting Graph of Total Mass of Fish Caught from Fish Farm
Part V
Model for New Data
Drawing Both Models
Part VI
Discussion on Trends of Both Models
Part VII
Possible Future Trends
Conclusion
Bibliography
Introduction
This portfolio considers commercial fishing in a country in two different environments, namely the sea and fish farms. The statistics are obtained from UN Statistics Division Common Database from the year 1980 to 2006. This task requires the plotting of graph and choosing a suitable model to represent the trends that appear in the graph plotted.
To develop the model trend for the data points, a number of methods and technology will be utilized. Two graphs will be plotted in this portfolio task. One is for fish caught in the sea and another is for the fish caught in fish farm. Towards the end of this portfolio task, it is required to discuss and evaluate the relationship between the two graphs.
Part I
Defining Variables, Parameters and Constraints
There are two variables in the data given. These two variables affect each other’s value. The independent variable is the year and the dependent value is the total mass of fish caught in the sea shown in thousands of tonnes. The change in year affect the total mass of fish caught.
Middle
Substitute values from data points:
Change into matrix form:
(Answer obtained using matrix solver in GDC)
The second equation,
Graph 4: Graph of Total Mass of Fish Caught In the Sea against Year 18 to 26
The graph was plotted using Graph software for Windows.
The data points are scattered but still in a linear line. Therefore, a linear function can be used to model this graph.
Linear function has the general formula:
To determine the parameters a and b, we will take two data points to be substituted in the equation.
(21, 527.7) and (23, 507.8)
Change into matrix form:
The last equation,
The mathematical model that can fit the data point is:
Part III
Model Function and Original Data Points
Graph 5: Graph of Total Mass of Fish Caught In the Sea against Year
The graph was plotted using GeoGebra graphing software.
From the graph plotted and the model function, it can be seen that the second function from the piecewise function fits the data accurately. The third function is also a good best fit line for the scattered points. However, the first function seems to produce a best fit line that does not follow closely the scattered points. This, perhaps result from the insufficient amount of data to derive the function as it only involves two points. Other points must be selected.
(1, 470.2) and (5, 575.4)
Change into matrix form:
The revised first equation,
Conclusion
The Code on Sea Fishing for the Future was developed in 1995 (Code on Sea Fishing for the Future, 2011) and this will be a barrier for open sea fishing and the decrease in total mass of fish caught in the open sea in the future.
The global trend of fish had been in decline since 1980 despite China’s inaccurate statistics (Fish Farming The Promise of a Blue Revolution, 2003) shows that global fish catch had decline and fish farming will replace open sea fishing.
To predict when will be the time the graphs intersect, equate the third function of first model and the second model:
From this calculation, in 2019, the fish caught from fish farm and the sea will be equal and from this year onwards, the fish caught from fish farm will be more than fish from the sea.
Conclusion
The model constructed had accurately represented the data points given and from the model, future values of total mass of fish caught can be predicted. From the prediction, the farm fish production will exceed the sea fish catch.
Bibliography
Fish Farming The Promise of a Blue Revolution. (2003, August 7). Retrieved October 27, 2011, from The Economist: http://www.economist.com/node/1974103
Code on Sea Fishing for the Future. (2011, October 14). Retrieved October 27, 2011, from Sea Fishing & Aquaculture: http://www.dpiw.tas.gov.au/inter,nsf/WebPages/ALIR-4YK7JE?open
Bard, Y. (1974). Nonlinear Parameter Estimation. New York: Academic Press.
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