# Math SL Fish Production IA

by deworst (student)

003717-011

Fish Production

I-Shou International School

Mathematics Standard Level Internal Assessment

Type 2

Candidate Name: Hung Li Chu

Candidate Number: 003717-011

Word Count: 2886

Date of Submission: 26th October 2012

The aim of this internal assessment is to consider commercial fishing in a particular country in two different environments- the sea and fish farms (aquaculture). The table of values below is taken from the UN Statistics Division Common Database.

Table 1: This shows the total mass of fish caught in the sea between 1980 and 2006 in thousands of tonnes.                                           (1 tonne = 1000 kilograms)

Graph 1: This shows the total mass of fish caught between the years 1980 and 2006.

Based on the graph, the x-axis represent the years between 1980-2006. The y-axis represents the total mass of fish caught in tonnes per year. The graph illustrates that in some years the total mass of fish caught increases whereas in other years the total mass of fish caught decreases. For instance, from 1980 until 1988, the total mass increases every year (426.8-669.9 tonnes). Then the total mass decreases from 1989 to 450.5 tonnes until 1992 where it increases again.

In order to develop a function model that fits the data points, the graph above will be split into 3 sections similar to how the table of values are split in table 1. The values for the years would have to be rearranged. This is done so that at 1980 x=0, and a y-intercept would occur. Therefore, at x=0, the y-intercept should be 426.8. By acquiring the x and y values through rearranging the values, it would be more convenient later when you substitute the values into the model function to find the coefficients.

Table 2: This shows the rearranged years of the 1st section starting with 1980 where x=0.

Graph 2: This is the 1st section of the original graph with the rearranged years.

Several function models were considered in order to find the most suitable one that would fit the first section of the graph. The linear function seems suitable because it behaves like the line of best fit, and since the points are fairly clustered together, a straight line could be drawn to connect the points together. However, a linear function does not have enough variables and since the graph does not have a direct linear relationship, a more suitable function would be a polynomial equation. A quartic function is suitable for this set of data because there are 5 variables in a quartic function. The more variables there are, the more accurate the function model would be at fitting the data points. Below is an example of a quartic function:

The x variable represents the year and the variable f(x) represents the total mass (tonnes). The parameters in this function is e, which is the y-intercept and the coefficients a, b, c and d are responsible for the curving of the graph.

In order to find the unknowns a, b, c and d, 4 years have been chosen (3, 4, 5 and 6) to substitute into x in the above function. These values have been chosen because they are consecutive years that lie in the middle of the graph. The y value would be substituted with the total mass of that corresponding year. The parameter e would be substituted with the y-intercept where x=0, y=426.8. Therefore 426.8 will be substituted into e.

Table 3: This shows the chosen values of the rearranged years, x, the total mass, y, and ...