• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month
Page
1. 1
1
2. 2
2
3. 3
3
4. 4
4
5. 5
5
6. 6
6
7. 7
7
8. 8
8
9. 9
9
10. 10
10
11. 11
11
12. 12
12
13. 13
13

# Population trends in China

Extracts from this document...

Introduction

MATHS COURSEWORK

POPULATION TRENDS IN CHINA

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 554.8 609 657.5 729.2 830.7 927.8 998.9 1070 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.

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

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

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

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

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

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))

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

## Found what you're looking for?

• Start learning 29% faster today
• 150,000+ documents available
• Just £6.99 a month

Not the one? Search for your essay title...
• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Related International Baccalaureate Maths essays

1. ## Math IA -Modelling Population Growth in China.

You first need to put all the data from the years 1950 to 1995 into L1 and the corresponding population data into L2. After you have put the data in the L1 and L2 lists. After the data has been entered into the lists.

2. ## Maths Portfolio - Population trends in China

approach is more detailed instead of the much broader one of the previous model. Points Value of the table (a) Value of the logistic curve (b) Difference between the values (b-a) Systematic error percentage ((b-a)/b)*100 50 554.8 546.8 -8 -1.5 55 609.0 611.3 2.3 0.4 60 657.5 680.4 22.9 3.4

1. ## A logistic model

These results are shown schematically and graphically: Table 4.1. Growth factor r=2.0 The population of fish in a lake over a time range of 20 years estimated using the logistic function model {5}. The interval of calculation is 1 year.

2. ## An investigation of different functions that best model the population of China.

to work out K, L and M. Given the five year intervals of the data, one can make "t" the number of years after 1950 and we already know that "P" means population. Therefore we have to input this into our GDC: x(t)

1. ## Finding Functions to Model Population trends in China

Therefore, I have the conclusion that linear function is not the model of the graph. Quadratic Function: y=ax2 +bx+c Quadratic function has three parameters, in which I will calculate them out below. I use (1955, 609), (1985, 1070) and (1960, 657.5)

2. ## Function that best models the population of China. Some of the functions that ...

A researcher suggests that the population, at time can be modelled by Where , , and are parameters. This is a logistic equation and the parameters in this equation each have a special role to perform: Parameters Role When the value of is increased the graph is shifted up by a certain number.

1. ## Population trends. The aim of this investigation is to find out more about different ...

The data given shows, as a starting point the year 1950, it then continues for another 45 years of data which has been given in multiples of 5. Year Population in millions 1950 554.8 1955 609.0 1960 657.5 1965 729.2 1970 830.7 1975 927.8 1980 998.9 1985 1070.0 1990 1155.3

2. ## In this Internal Assessment, functions that best model the population of China from 1950-1995 ...

A quadratic fit applied to the data points via Graphical Analysis 3 follows: Cubic Fit: From the graph, the data points also seem to show trends apparent in graphs of cubic functions, as the rate at which the population increases proliferates as the years go by.

• Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to