- Level: International Baccalaureate
- Subject: Maths
- Word count: 3743
The Population of Japan and Swaziland
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Introduction
Rachel Timmons
The Population of
Japan and Swaziland
Type 2 Portfolio
Rachel Timmons
Lee’s Summit West High School
IB Mathematics SL
5/18/2009
Population models are formulas that one can use to calculate the future population of a country based on past growth. These growths can sometimes be shown exponentially. In this portfolio, I will be finding population models for the countries of Swaziland and Japan.
I will begin with Swaziland. Using the data in the following table, I will find an exponential function algebraically to describe the population based on the year. A possible format for this function is, and this is the one I’ll be basing my model on. The following data was taken from www.library.uu.nl/wesp/populstat/afica/swazilac.htm . The populations shown are estimates.
Year | Population (thousand) | Year | Population (thousand) |
1911 | 100.0 | 1960 | 330.0 |
1921 | 112.8 | 1970 | 422.0 |
1927 | 122.0 | 1980 | 565.0 |
1936 | 156.7 | 1990 | 751.0 |
1944 | 171.3 | 2000 | 1083.3 |
1950 | 264.0 | 2005 | 1317.0 |
In order to make the data easier to work with, I’m going simplify the years according to 1911 being year 1.Therefore, year 1921 will be represented as year 11, 1927 as 17, 1936 as 26, etc. These new expressive values will be used as my x values and the population as my y values. I placed these values in the table below.
X | Y | X | Y |
1 | 100.0 | 50 | 330.0 |
11 | 112.8 | 60 | 422.0 |
17 | 122.0 | 70 | 565.0 |
26 | 156.7 | 80 | 751.0 |
34 | 171.3 | 90 | 1083.3 |
40 | 264.0 | 95 | 1317.0 |
My process will be to first find the rate, b. I am going to obtain this value by finding the difference in population divided by the change in years. Then, I’ll take that quotient and add it to the smaller of the two populations; that sum will then be divided by that same smaller population. I will do multiple trials of this process using only data adjacent to each other. The results of these trials will then be averaged. I’ll demonstrate this process using the first two consecutive values in the data table with the use of my TI-84 Plus Silver Edition Graphics Display Calculator (GDC).
Middle
If x=110 then y≈1964.496 (thousand). Therefore, the possible population of Swaziland in 2020 is 1,964,496 people.
If x=120 then y≈2640.118 (thousand). Thus, the estimated population of Swaziland in 2030 is 2,640,118 people.
According to the CIA Factbook https://www.cia.gov/library/publications/the-world-factbook/geos/wz.html#People, the estimated 2009 population is expected to be 1,123,913 people. Compared to my prediction of the year 2010 having the population 1,461,769 people and having the record of 2005 having the population approximately 1,317,000 people, this estimation can elicit a conclusion that the population of Swaziland is actually shrinking instead of growing as my population model portends. This shrinking could be a result of disease such as AIDS, or other environmental conditions.
I am now going to proceed in finding a population model for Japan using the same methods and ideas I did for Swaziland. I will again use the exponential function, , and find each variable using algebraic processes. The following table shows approximate populations measured in millions of given years.
Year | Population (million) | Year | Population (million ) |
1900 | 43.8 | 1960 | 93.4 |
1910 | 49.6 | 1970 | 103.7 |
1920 | 56.0 | 1980 | 117.1 |
1930 | 64.5 | 1990 | 123.5 |
1940 | 71.9 | 2000 | 123.9 |
1950 | 83.2 | 2005 | 127.7 |
I reused the same idea of simplifying the years that I did when finding the model for Swaziland. I started with 1900 as the first year, 1910 as the eleventh, 1920 as the twenty-first, and so on. These simplified values will express my x value and the years (in millions) will express my y values. I placed these values in the table below.
X | Y | X | Y |
1 | 43.8 | 61 | 93.4 |
11 | 49.6 | 71 | 103.7 |
21 | 56.0 | 81 | 117.1 |
31 | 64.5 | 91 | 123.5 |
41 | 71.9 | 101 | 123.9 |
51 | 83.2 | 106 | 127.7 |
I am going to repeat the same procedures I used to find the model for Swaziland in finding the model for Japan. Therefore, I will first find the rate, b. To reiterate what I did in finding this value for Swaziland, I took the change in population divided by the change in year. I then added that quotient to the smaller of the two populations and divided that sum by the same smaller population.
Conclusion
If t=130 then P(t)≈135.996. Thus, in 2030 the population of Japan is expected to be near 135,996,000.
I think these predictions are more probable. They fit with the trend of the data more closely, as could be observed by the graph I made with my GDC.
The last three points on this graph are data predicted from my model. As can be observed, the graph of the model follows the pattern of the seeming “leveling off”. This results in the predictions being more feasible.
These estimates fit better with the projections from CIA Factbook, which stated that the estimate 2009 population would be 127,078,679. Compared to the 2010 projections from my model, it is fairly close. I believe this form of model is more preferable because of its having a capacity, which I had mentioned earlier as a possible part of the reason Japan’s population appears to be leveling off.
Another way I can check for accuracy of predictability in this model would be to use it to calculate population estimates for past years and compare them to the actual recorded populations. I will demonstrate this below using my GDC with year 30.
As listed in earlier pages, the actual population in year 30 was about 64.5 million. According to my model, the population should be approximately 69.1 million. As you can see, the population values are similar but not equivalent. This is a representation of how accurate is my model’s ability to predict. Although it is not a perfect model, it is able to calculate reasonable results that can assist in population estimates. Overall, I think the logistic model is a better fit for determining estimations of future populations.
Works Cited
“Japan.” CIA - The World Factbook. 23 Apr. 2009. 13 May 2009 <https://www.cia.gov/library/publications/the-world-factbook/geos/ja.html>.
“Swaziland.” CIA - The World Factbook. 23 Apr. 2009. 13 May 2009 <https://www.cia.gov/library/publications/the-world-factbook/geos/wz.html>.
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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