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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

Extracts from this document...

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.

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Middle

Substitute values from data points:

image08.png

image09.png

image10.png

image11.png

Change into matrix form:

image12.png

image13.png

image15.png

(Answer obtained using matrix solver in GDC)

The second equation,

image16.png

Graph 4: Graph of Total Mass of Fish Caught In the Sea against Year 18 to 26

image17.jpg

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:

image18.png

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)

image19.png

image20.png

Change into matrix form:

image21.png

image22.png

image23.png

The last equation,

image24.png

The mathematical model that can fit the data point is:image00.png

image26.png


Part III

Model Function and Original Data Points

Graph 5: Graph of Total Mass of Fish Caught In the Sea against Year

image27.png

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)

image28.png

image29.png

Change into matrix form:

image30.png

image31.png

image32.png

The revised first equation,

image33.png

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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:

image53.png

image54.png

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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