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Population trends in China

Extracts from this document...

Introduction

Claudia Nevado        003656-008

MATHS COURSEWORK

POPULATION TRENDS IN CHINA

Claudia Nevado

INDEX

1. Finding my own model(s)

2. Research model using my own method and find K, L and M

3. Input new data using previous model and research model.

  1. The aim in this coursework is to investigate the different functions that best model the population of China from 1950 to 1995.

The following table shows the population of China between these years:

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

The relevant variables in this investigation are the population in millions in different years. The parameter is the initial population growth.

The data points from the table above are shown in the graph below, using Microsoft Excel showing the population in China from 1950 to 1995 (presenting the years from 0 being 1950, to 45 being 1995):

What I can observe according to the graph beside is that as the years pass, the population of China (in millions) increases gradually. image00.png

The functions which could model the behaviour of the graph can be any of the following:

· y = mx+c

The graph appears linear therefore we could use the

...read more.

Middle

14.5

Years

Year

Population in Millions

Model 3

 1950

0

554.8

554.8

1955

5

609

627.3

1960

10

657.5

699.8

1965

15

729.2

772.3

1970

20

830.7

844.8

1975

25

927.8

917.3

1980

30

998.9

989.8

1985

35

1070

1062.3

1990

40

1155.3

1134.8

1995

45

1220.5

1207.3

image03.png

2. A research suggests that the population, P at time t can be modeled by: P (t) =       K /1+Le -Mt        where K, L and M are parameters.

Firstly, to find what K is I will place 0 into t (time).

Through mathematical knowledge we may say that e –Mt will equal to 1, because when t=0 M will also be 0 when multiplied together, and any number to the power of 0 will always equal to 1.

When t=0, the population is 554.8, meaning the equation would be:

    554.8 = K / 1+ L (*1)

If we say L = 1, then K must equal 2*554.8, which is 1109.6

Being the final equation: P (t) = 1109.6 / 1+ L

By using Microsoft Excel I tried estimating the possible values of K, L and M by entering the data and plotting graphs.

Being:

K

L

M

1109,6

1

0,06

Years

Year

Population in Millions

Research Model

1950

0

554,8

554,8

1955

5

609

637,4

1960

10

657,5

716,4

1965

15

729,2

788,9

1970

20

830,7

852,8

1975

25

927,8

907,2

1980

30

998,9

952,2

1985

35

1070

988,5

1990

40

1155,3

1017,3

1995

45

1220,5

1039,7

image04.png

The graph shows that the research model line does not fit the actual data; therefore I will try other numbers for K, L and M so that it becomes more accurate.
I changed my constants using trial and error and finally got a model which fits better:

K

L

M

1600

2

0,04

Years

Year

Population in Millions

Research Model

1950

0

554,8

533,3

1955

5

609

606,6

1960

10

657,5

683,6

1965

15

729,2

762,8

1970

20

830,7

842,7

1975

25

927,8

921,8

1980

30

998,9

998,5

1985

35

1070

1071,5

1990

40

1155,3

1139,8

1995

45

1220,5

1202,5

image05.png

The model would be: 1600/1+e-0.04t.

In China, families by law are allowed to have a maximum of one child per family only; therefore this could explain why this population trend tends to be almost linear.

  1. Another table was given to me with the population in China in millions from year 1983 to 2008:

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

...read more.

Conclusion

class="c7">9

1171,7

1160,6

1252,4

1997

14

1236,3

1233,1

1400,3

2000

17

1267,4

1276,6

1490,1

2003

20

1292,3

1320,1

1580,1

2005

22

1307,6

1349,1

1639,7

2008

25

1327,7

1392,6

1728,4

I finally modified my final model and the research model so that they could fit the IMF data as accurately as possible, which applies to the given data from 1950 to 2008.

Years

Years

Population in Millions

My model 3

Research Model

1950

0

554.8

554.8

533.3

1955

5

609

627.3

606.6

1960

10

657.5

699.8

683.6

1965

15

729.2

772.3

762.8

1970

20

830.7

844.8

842.7

1975

25

927.8

917.3

921.8

1980

30

998.9

989.8

998.5

1983

33

1030.1

1033.3

1042.8

1985

35

1070

1062.3

1071.5

1990

40

1155.3

1134.8

1139.8

1992

42

1171.7

1163.8

1165.5

1995

45

1220.5

1207.3

1202.5

1997

47

1236.3

1236.3

1225.9

2000

50

1267.4

1279.8

1259.2

2003

53

1292.3

1323.3

1290.3

2005

55

1307.6

1352.3

1309.8

2008

58

1327.7

1395.8

1337.2

The graph shown more underneath, demostrates how my final model and the research model have been modified so that they can slightly fit the IMF data. As you can see, I finally obtained a very precise data which fits the actual line thorugh trial and error.

Below is also shown how I modified my models, where you can see the formula used for each one in relation to all the IMF data.

Years

Years

Population in Millions

My model 3

Research Model

1950

0

554.8

=14.5*B2+$C$2

=1600/(1+2*2.71828^(-0.04*B2))

1955

5

609

=14.5*B3+$C$2

=1600/(1+2*2.71828^(-0.04*B3))

1960

10

657.5

=14.5*B4+$C$2

=1600/(1+2*2.71828^(-0.04*B4))

1965

15

729.2

=14.5*B5+$C$2

=1600/(1+2*2.71828^(-0.04*B5))

1970

20

830.7

=14.5*B6+$C$2

=1600/(1+2*2.71828^(-0.04*B6))

1975

25

927.8

=14.5*B7+$C$2

=1600/(1+2*2.71828^(-0.04*B7))

1980

30

998.9

=14.5*B8+$C$2

=1600/(1+2*2.71828^(-0.04*B8))

1983

33

1030.1

=14.5*B9+$C$2

=1600/(1+2*2.71828^(-0.04*B9))

1985

35

1070

=14.5*B10+$C$2

=1600/(1+2*2.71828^(-0.04*B10))

1990

40

1155.3

=14.5*B11+$C$2

=1600/(1+2*2.71828^(-0.04*B11))

1992

42

1171.7

=14.5*B12+$C$2

=1600/(1+2*2.71828^(-0.04*B12))

1995

45

1220.5

=14.5*B13+$C$2

=1600/(1+2*2.71828^(-0.04*B13))

1997

47

1236.3

=14.5*B14+$C$2

=1600/(1+2*2.71828^(-0.04*B14))

2000

50

1267.4

=14.5*B15+$C$2

=1600/(1+2*2.71828^(-0.04*B15))

2003

53

1292.3

=14.5*B16+$C$2

=1600/(1+2*2.71828^(-0.04*B16))

2005

55

1307.6

=14.5*B17+$C$2

=1600/(1+2*2.71828^(-0.04*B17))

2008

58

1327.7

=14.5*B18+$C$2

=1600/(1+2*2.71828^(-0.04*B18))

image07.png

...read more.

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