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Mathematical Modeling

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Introduction

Year 12 Mathematics Standard Level

Mr. A. Mumm

Medora Choi

8th March, 2010

Mathematics SL Portfolio: Type II

China’s Population Growth

        Mathematics is a study of the concepts of quantity, structure, space and change. It is a type of science that draws conclusions and connections to the world around us. Mathematicians would call math a science of patterns and these patterns are discovered in numbers, space, science, computers, imaginary abstractions, and everywhere else. Mathematics is also found in numerous natural phenomena’s that occurs around us. Today math is used all around us and is applied to many educational fields, through this people have become inspired to discover and make use of their mathematical knowledge which will then lead to entirely new disciplines. Math is present in wherever there are difficult problems that involve quantity, structure, space or change; such problems appear in various forms such as commerce, land measurement and especially astronomy.

The purpose of this paper is to examine the different functions and investigate the best model for the population of China from 1950 to 1995. The following table shows the population in million of China during the year of 1950 to 1995:

Year

1950

1955

1960

1965

1970

1975

1980

1985

1990

1995

Population in Million

554.8

609.0

657.5

729.2

830.7

927.8

998.9

1070.0

1155.3

1220.5

...read more.

Middle

To test out this function, I have to choose two points from the data to find out the gradient of the function,

First Point: (0, 554.8)

Second Point: (5, 609)

To find the gradient, I use the formula

image17.png

Therefore it is shown as:

m =image18.png

The next step is to find b in the function, I can find this value by inputting a data point into the function like this:

image19.png

Thus, generating the complete linear function:

image20.png

I then input this function into the graph to test the accuracy of the hand-calculated function.

image21.png

As seen from the graph above, the line does not show the best fit as it does not touch upon all the lines, suggesting that the hand-calculated function may not be entirely reliable.

I then used my GDC to find the value for m and b, which is:

image03.png

I then plot it into GeoGebra and it fits the line perfectly as shown as below:

image04.png

To find a better fit line for the graph, I used a [FitLine] function from the GeoGebra to find a more accurate function, which looks like this:

image04.png

This shows that the GDC and the GeoGebra are much more reliable resources than hand-calculated.

...read more.

Conclusion

         To further compare the two models, there are additional data on population trends in China from the 2008 World Economic Outlook, published by the International Monetary Fund (IMF) which is shown on the table below:

Year

1983

1992

1997

2000

2003

2005

2008

Population in Millions

1030.1

1171.7

1236.3

1267.4

1292.3

1307.6

1327.7

The above data is individually tested on each of the models. For the first model, which is also the linear model, has only three points touching the model, including year 1983, 1992 and 1997; the rest of the data points were lower than the model. The second model is then tested, and for this researcher’s model, all the points are extremely close are touching the model, meaning the modified model should be based on the researcher’s model as it is more reliable than the linear model.

Hence, this model is established:

image13.png

This model fit all the IMF data; hence it applies to all the given data from year 1950 to 2008 where the maximum points can touch on this model as shown below:

image14.png

The modified model shows that the general data points are on the line, meaning this is the best that fits all the data since 1950 until 2008.

...read more.

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

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