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Finding Functions to Model Population trends in China

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Introduction

Math Portfolio II

Population Treads In China

By Amber Perng

Aim: In this task, you will investigate different functions that best functions that best model the population of China from 1950 to 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

By plotting the above data points in autograph, we get:

image00.png

Graph 1: Population of China 1950~1995

Variable

x-axis: the time (t)

y-axis: the population of China (p)

The above graph is plotted according to the population of China from 1955 to 1995. The points show that the population has increased at a constant rate, but had increased a little more from 1765 to 1970 comparing to other years, before it returned back to the constant increase.

To create a model function to fit the behaviour of this graph, I will use linear,

quadratic and exponential  functions to find its best fit.

Linear Function: y=ax+b

First, I use two points of (1955, 609) and (1985, 1070) from the given data to find the parameters of the function.

I chose these two points because they are the only two combinations that are all integers, which will be easier to calculate.

609=1955a+b

...read more.

Middle

image02.pngimage02.png= image03.pngimage03.png using Graphic Digital Calculator

a≒0.23

b≒-877.70

c≒850186.83                                      

Which gives the equation of y=0.23x2-877.7x+850186.83, as below:

image04.png

Graph 3: Population of China 1950~1995 with quadratic function

According to the graph, we can see that quadratic function did not fit the points at all. Below is a recalculation by plotting variable x into the equation:

0.23(1955)2-877.7(1955)+850186.83= 13349.08  → off by 12740.08

0.23(1960)2-877.7(1960)+850186.83= 13462.83  → off by 12895.33

0.23(1985)2-877.7(1985)+850186.83= 14024.08  → off by 12954.08

From the recalculation, we could see that this equation has an inconstant parameter c, which it varies the value of y when adding itself onto the number, thereby unable to provide an appropriate curve.

I then used Graphic Digital Calculator to revise the equation, below is what the values given by the calculator:

y=ax2+bx+c

a=0.355454545

b=-124.7309394

c=108601.1173

It shows that the values of what I calculated are far from the correct value; this may be another reason of the result of the incorrect graph.

Quadratic Function II: y=a(x-b)2+c

I then decided to try another type of quadratic function.

...read more.

Conclusion

A researcher suggests that the population, P at time t can be modeled by:

image14.pngimage14.png , where K, L and M are parameters.

This is a simple logistic function.

When t=0, P(0)=image15.pngimage15.png , which K is at its maximum because it is indirectly proportional to parameter t.

Therefore, according to the given data from the above table, we can use GDC calculator to find the parameters:

K≒1946.18

L≒4.34

M≒0.33

By substituting the values of K, L and M into the formula, we have:

image16.png

image17.png

Graph 5: Population of China in logistic function

Here are additional data on population trends in China from the 2008 World Economic Outlook, published by the International Monetary Fund (IMF):

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

In which we have the following graph containing all the points:

image18.png

Graph 7: The final population of China graph

I use the function image14.pngimage14.png and find the parameters using Graphic Digital Calculator:

K≒1617.46

L≒1.35

M≒0.399

Which I then plot the equation into the graph:

...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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