 Level: International Baccalaureate
 Subject: Maths
 Word count: 1050
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: 002603011
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 
Middle
10
657.2
15
729.2
20
830.7
25
927.8
30
998.9
35
1070
40
1155.3
45
1220.5




Matrix:
Input looks like this:
I will now use the rref() function on my TI84 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 
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
TI84; 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 TI84 (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]
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