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IB Math SL Portfolio Type 2 Population in China

Extracts from this document...

Introduction

IB Mathematics SL

Math Portfolio

(Type 2)

Population Trends in China

Candidate Name: Ishaan Khanna

Candidate Session Number: 002603-011

Session: May 2012

The Doon School

Index

Introduction

Variables

Parameters

Modelling the population of China

Linear

Quadratic

Cubic

Finding Equation analytically

Researcher’s equation

Implications

Cubic model

Researcher’s model

Additional data

Cubic model

Researcher’s model

Final model

Original Data

Conclusion

Bibliography

Introduction

The aim in this portfolio is to study the different functions that best model the Chinese population from 1950 to 1995.

The following table shows the population of China between 1950 and 1995:

Year

1950

1955

1960

1965

1970

1975

1980

1985

1990

1995

Population in Millions

554.8

609.0

657.5

729.2

830.7

927.8

998.9

1070.0

1155.3

1220.5

Variables

The significant variables in this analysis are ‘x’ and ‘y’. The variable ‘x’ is for the different years from 1950 to 1995 with 1950 as 0. The population in millions is represented by the variable ‘y’.

Parameters

The variables are time and population. Time (represented by x) cannot be negative and cannot decrease because it represents a unit of time. It can however increase infinitely. The variable population (represented by y) cannot be negative as population cannot go below 0. In this case the population also cannot increase infinitely as China will not be able to handle so many people. (Maximum carrying capacity of a country)

I will now list the values of ‘x’ and ‘y’.

X Value

Y Value

1950

554.8

1955

609

1960

657.2

1965

729.2

1970

830.7

1975

927.8

1980

998.9

1985

1070

1990

1155.3

1995

1220.5

...read more.

Middle

10

657.2

15

729.2

20

830.7

25

927.8

30

998.9

35

1070

40

1155.3

45

1220.5

0

554.8

15

729.2

30

998.9

45

1220.5

Matrix:  

Input looks like this:

I will now use the rref() function on my TI-84 to solve the matrix and get a single equation:

On pressing I receive:

The output is:

Equation is:

Therefore the equation I get is:

The Graph below shows the equation (the red line is the data set):

Since the equation I developed is not representing the data set well enough (Graph 5) I will use the cubic equation I had got earlier via technology.

Researcher’s equation

A researcher suggests that

This is a logistic function where K,L and M  are the parameters.

Using Logger Pro, I input the equation and use the ‘Best Fit’ function. I receive:

In this case:

The graph of the equation is shown below:

In graph 7 I will compare the researcher’s function against the original data:

As we can see the researcher’s model does not fit the data as well as the cubic function.

Implications

I will now see what the 2 models predict the future populations will be:

Cubic model

Researcher’s model

As we can see, the cubic model (graph 7) is declining which is not possible for a country’s population. Thus the cubic equation cannot be relied upon.  The Researcher’s model (graph 8) shows exponential growth which is more realistic.

Additional data

I will now compare the two models with the new information I have received and see how well they have predicted the values.

Year

Population in Millions

1950

554.8

1955

609

1960

657.5

1965

729.2

1970

830.7

1975

927.8

1980

998.9

1983

1030.1

1985

1070

1990

1155.3

1992

1171.7

1995

1220.5

1997

1236.3

2000

1267.4

2003

1292.3

2005

1307.6

2008

1327.7

...read more.

Conclusion

The new equation is:

Its future implications:

Conclusion

With the new data the Researcher’s Equation is much better. The modified model forms an ‘S’ shaped logistic function which is best suited to show population growth. Some of the points in the beginning of the graph are a little off and the last few points show a decrease in the rate of population growth. However, it does model the general increase and the levelling off of the population and gives a decent estimate of the population increase for future years.

According to me  is the best function to forecast China’s population.

Bibliography

TI-84; Solving Linear Equations with Row Reductions to Echelon Form on Augmented Matrices method learnt from:

http://education.ti.com/xchange/US/Math/PrecalculusTrig/8838/Precalculus_RREFonSystems_Trogdon.pdf

[Accessed on 21 February 2011]

Technology used:

Texas Instruments 84 Silver Edition calculator

Microsoft Paint (Version 6.1)

MathType (Version 6.7)

Microsoft Word 2010

Vernier Logger Pro

Emulator Used: Virtual TI-84 (Wabbitemu by Revolution Software)

Graph 1

Graph 2

Graph 3

Graph 4

Graph 5

Graph 7

Graph 6

Graph 8

Graph 9

Graph 10

Graph 11

Graph 12

Graph 14


[1] Logger pro

[2] Method learnt from:

http://education.ti.com/xchange/US/Math/PrecalculusTrig/8838/Precalculus_RREFonSystems_Trogdon.pdf

[Accessed on 21 February 2011]

...read more.

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