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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

Cubic

Finding Equation analytically

Researcher’s equation

Implications

Cubic model

Researcher’s model

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 554.8 609 657.5 729.2 830.7 927.8 998.9 1070 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

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:

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:

## 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.

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

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

 Logger pro

 Method learnt from:

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

[Accessed on 21 February 2011]

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

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