# Mathematics internal assessment type II- Fish production

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Introduction

Introduction

The aim of this task investigates the activity of commercial fishing in a certain country in the sea environment, as well as fish farms. The sets of data from both fish caught in the sea, as well as fish from the fish farms. This data was obtained from the UN statistics Division Common Database.

The main purpose of this task is to define the variables and parameters/constraints of the data given, and using technology, plot the data points into a graph, and develop, and discuss a model for the different data points for both environments that relate to their trends

Different types of software will be utilized to develop the models for each set of data points for both the environments, accompanied by two separate graphs for each type of environment. After the functions and models are made, they will be used to describe the current trends observed by each environment, along with possible future trends.

Definition of variables and parameters/constraints

Before we can plot the data points into a graph, we must first define the variables from the data given, as well as discuss any parameters. There are two different variables given to us, namely the independent and dependent variables. The different variables in the given data are as follows:

- Independent: The independent variable would be the year. This variable is changed by increasing the year by one, and taking note of the total mass of fish caught in that particular month.
- Dependent: The dependent variable for this set of data would be the amount of fish caught in each year. This variable is observed by noticing the mass of fish caught after every incremental change to the year. The mass of fish caught can only change when the year has changed, which is why it is dependent on the change of years

The parameters or constraints in a particular set of data points are the factors in the mathematical model that are constants related to the independent and dependent variables. They are also known as constant variables in a set of data points. Possible parameters may be environmental factors such as the weather, economical factors, government policies, demands of the fish in local and international markets and so on.

Plotting of points (Fish caught in the sea)

The data points were plotted using GeoGebra 4 for Mac OSX, and the x values have been substituted for a better fit of the graph. The points have been plotted into the scatter point graph below:

Trends found in the graph

Upon first glance of the graph, we can notice a number of trends and characteristics of the graph. From the years 1980 to 1988, or in terms of the graph, until the 8th point of the x-axis, there is a linear increase or increment observed in the graph, in fact, we can see that the entire graph reaches its highest point in 1988, at approximately 670 tonnes of fish caught. Furthermore, the number of fish caught tends to reduce, thus resulting in a lesser total mass of fish caught, hence the graph depicting a downward slope. This continues until the year 1991, where the graph is at its lowest point, at approximately 357 tonnes of fish caught. After the year 1991, the graph displays another upward slope all the way to the year 1995, where the graph reaches its second highest point, at approximately 634 tonnes of fish caught. After the year 1995 all the way to 2006, the graph fluctuates and rises and falls thereby providing a wave-like form. Overall, the only decrease we can see in the graph is between the years 1998 and 1991.

Suggestion of models

According to the trends in the graph that were observed, a number of models or functions may be associated with a particular set of points in the graph, which results in a number of models that can be suggested for the graph such as:

- Linear function: The graph displays a constant or linear incremental increase in a number of areas, and as a whole, the graph also appears to be increasing as the total mass of fish increases every year with the exception of decreasing sometimes. Therefore, a linear model can be used to describe the graph
- Cubic function: The graph, particularly from the year 1988, illustrates the shape of a cubic graph or what the graph of a cubic function may appear to look like, as it shows a upward and downward sloping shape of the graph, which may be related to a cubic function, hence the graph can also be interpreted by a cubic function or model
- Polynomial function: A polynomial graph is generally represented by an alternating increase and decrease of the graph, which this graph in particular illustrates a great amount, as a polynomial graph is best described as a collection of “upward” and “downward” curves, hence it is the best way to describe this graph.

Middle

The equation for this function is depicted as:

fx= ax+b

Therefore from this equation, there are 2 unknown variables, namely a and b. To define these variables, we can take two points from the graph.

The two points that are taken from the graph would be (2, 470.2) and (6, 575.4)

Hence substituting the values of a and b:

2a+b=470.2

4a+b=575.4

Thus, solving the equations using the Polysimul program in the Graphic Display Calculator (GDC):

Hence forming the equation for the first interval:

fx= 52.6x+365

Here is line of the equation against the interval of the graph:

As we can see, the line does not quite match the graph in this particular interval, due to the incorrect start position for the graph. Hence the value of 1980 will be substituted with 0, 1981 with 1, and so on. Hence the new graph with the shifted values would be like so:

Therefore the new values would be (1, 470.2) and (5, 575.4)

Hence substituting the values of a and b:

a+b=470.2

5a+b=575.4

And solving the equation using the PolySimul:

Hence forming the equation:

fx= 26x+444

And here is the line of the equation plotted against the graph:

As we can see, the line fits the graph perfectly.

Graph for interval 1989 to 1998:

This graph illustrates a upward and downward curve trend that can be described by a polynomial function.

Conclusion

Finally, advancing technology in favour of fish farms could also be vital in the increase of fish from the fish farms, due to increased variety and improved quality from the fish in the farms, which is also another likely cause for the exponential rise of amount of fish in fish farms.

Possible Future trends

While a gradual decrease is noticed from fish from the sea, it tends to show an upward linear trend towards the end of the graph, as opposed to the continuous display of an exponential trend in the fish from the farms. Possible future trends would be a direct continuation of each of the trends from both of the environments. However, due to its exponential increase, the fish from the fish farms will catch up with the fish from the sea, and the two graphs will intersect, in turn causing the farm fish to replace the sea fish as the main supplier of fish. To determine the exact point in time in the graphs where they will intersect, the functions from the third interval from each model suggested for both data points can be equated, giving an x value of 40, which is the point in time where the two graphs will intersect, which is approximately in the year 2020. This break-even point will cause the fish from the fish farm to replace the fish from the sea as the main contributor or supplier of fish for the many years to come.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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