CHINA POPULATION IB Pre-Calculus Period 6 November Alex Borowski
Introduction: Introduction of China Population trend
In this investigation, I will be discussing the trend within the population growth of Chinas population during the years of 1950 to 1995. Populations tend to rise at ever-increasing rates, but soon start to slow down, eventually leading into a sustainable population size. Chine is one of the best examples for this trend.
Data: Population of China from 1950 to 1995 Year (to accommo date analytical work)
0
5
10
15
20
25
30
35
40
45
Year
195 0 554 .8
195 5 609 .0
196 0 657 .5
196 5 729 .2
197 0 830 .7
197 5 927 .8
198 0 998 .9
198 5 107 0.0
199 0 115 5.3
199 5 122 0.5
Populati on in Millions
X is the year that the population was present. Y is the population in millions during that present year.
Parameters This chart shows the association in the middle of Years on account of 1950 and Population in Millions of China. The X pivot is the Years seeing that 1950 (with 5 year intervals), while the Y pivot is Population in Millions. Taking a gander at the diagram, one can perceive that at every 5 year interval, the population in millions builds exponentially until 30 years following 1950 (1980) where the expand slowly diminishes until it touches a horizontal asymptote where it would be able to never go above. The parameters is set at 0
For this chart, there are a couple of functions that could pose the development of this diagram. For example, an exponential role itself y a(b)x would pose the method on the grounds that much the same as the diagram, the exponential method increments by a ratio. Nonetheless, the finish conduct of the Chinese population diagram (as x infinity, f(x) h orizontal asymptote) is not similar to the exponential method for the reason that it bit by bit diminishes its rate of increment (ratio) until it goes at a horizontal asymptote, while the exponential role continually builds. A preferred function that would be able to fit the Chinese Population chart may be the logistic function. Not just does it increment exponentially (as does the chart above), but it likewise diminishes its ratio until it is going at a horizontal asymptote.
Finding Functions to represent the model: For this graph, there are a couple of functions that may model the growth of this graph. For instance, an exponential function itself [y=a(b) x] may model the function because just like the graph, the exponential function increases by a ratio. However, the end behavior of the Chinese population graph (as x--> infinity, f(x)---> horizontal asymptote) is not like the exponential function because it gradually decreases its rate of increase (ratio) until it approaches a horizontal asymptote, while the exponential function constantly increases. A better function that could fit the Chinese Population graph can be the logistic function . Not only does it increase exponentially (as does the graph above), but it also decreases its ratio until it is approaching a horizontal asymptote.
Basic Logistic Function y= (c)/1+abx
However, you can ignore Quadrant 2 of the model because it extends out of the parameters Calculating an equation to fit the graph: 1. The equation to find the Logistic Function y= (c)/1+abx y: Population in millions c: Limit of growth (horizontal asymptote) a: Constant: determined by initial value b: growth factor (ratio) x: years (since 1950) 2. Substitute the variables with the known values. For the values of x and y, you can use any data value point, but the most logical choice would to start with the x being zero
554.8=(1900)/1+ab0 P=(1900)/1+abx 554.8=(1900)/1+ab0 (1+ab0)554.8=(1900)/1+ab0 x 1+ab0 554.8 + 554.8a=1900 554.8+554.8a=1900 -554.8 -554.8 554.8a=1345.2 /554.8 /554.8 a= 2.42 P=(1900)/1+2.42bx 830.7=(1900)/1+2.42b20 (1+2.42b20)830.7=(1900)/1+2.42b20 x1+2.42b20 830.7 + 2010.3b 20=1900 830.7+2010.3b20=1900
-830.7
-830.7
2010.3b20=1069.3 /2010.3 /2010.3 20th root(b 20)=20th root(.532) b=0.97 Y= 1900/1+2.42(0.97)x
Comparing Chinas population of original data & the Function y= (1900)/1+2.42 (.97)x
Year Populatio n in Millions Population growth from equation y= (1900)/1+2 .42 (.97) x
195 0 554 .8
19 55 60 9.0
19 60 65 7.5
196 5 729 .2
19 70 83 0.7
19 75 92 7.8
19 80 99 8.9
198 5 107 0.0
199 0 115 5.3
199 5 122 0.5
555 .56
617. 26
682. 33
750 .26
820. 39
891. 99
964. 25
1036 .4
1107. 5
1176 .8
Difference in population from original data to equation y= (1900)/1+2. 42(.97)x
.76
8.26
24.8 3
21.0 6
10.3 1`
35.8 1
34.6 5
-33.6
-47.8
-43.7
My logistic function as examined in relation to the given information of Chinese Population associate truly nearly, in particular at the early years following 1950. They both are on the same parameters (0 x 50), and have the same exponential development from the start, then afterward begin to show at least a bit of kindness ratio until they are going at a horizontal asymptote until infinity. Notwithstanding, it is not flawless to the information given. The close conduct of the reasonable domain is easier than that of the given data's, and that is one great issue. Researchers suggested modeled equation for China population
P(t)= K/(1+Le-Mt) Using Technology to find the modeled equation 1. With the model P(t)= (K)/1+Le-Mt–, I input all of the original data given 2. Compute the logistic regression K=1946, Limit of growth (horizontal asymptote) L=2.61, Constant M=0.033, growth factor 1946/(1+2.61e-.033t)
P(t)=
Chinese Population Over the Years 1400 1200 s n o il 1000 il m 800 in ( n o ti 600 a l u p o 400 P
Population in Millions Researcher's Model
200 0 0
10
20
30
Years since 1950
40
50
Comparing Chinas population growth of the original data to the equation P(t)= 1946/(1+2.61e -0.033t)
Year Populatio n in Millions Population growth from equation P(t)=1946/ (1+2.61e0.033t ) Difference in population from original data to equation P(t)=1946/ (1+2.61e0.033t )
195 0 554. 8
195 5 609. 0
196 0 657. 5
196 5 729 .2
197 0 830 .7
197 5 927. 8
19 80 99 8.9
537. 62
604. 13
674. 92
749 .38
826 .71
905. 98
986 .15
17.1 8
-4.87
17.4 2
20. 18
3.9 9
21.8 19
12. 75
198 5 107 0.0
199 0 115 5.3
106 6.1
114 4.9
3.90 0
10.3 99
199 5 122 0.5 122 1.4
.9
No matter how far they contrast in terms of information (admitting that the first information, my logistic role, and the analyst's pose capacity all look at correspondingly), the stated models all demonstrate that China has grim work to revisit to in terms of population development. In simply 50 abbreviated years, the nation's population rose up to roughly twice its sum. Although the aforementioned models predict that China will in the future level off its population development (demonstrated by the point of confinement of development in all poses), it nevertheless will have awfully numerous folks to manage itself pleasingly. China Population growth from 2008 World Economic Outlook, published by th e International Monetary Fund (IMF)
Year
1983
1992
1997
2000
2003
2005
2008
Population in Millions
1030. 1171. 1236. 1267. 1292. 1307. 1327 1 1 3 4 3 6 .7
The first information, when looked at in relation to the IMF information given, in fact reliably comapres. For example, the IMF information for 1983 is 1030. 1 million individuals, while the 1980 information for the first given is simply 30 million individuals off (at 998. 9 million). This is steady with the other years (1995 from the first and 1993 from the IMF). What one should consider, notwithstanding, is that there are a couple of year departures from every of the information focuses. In this manner, considering the stated, the first information given can reliably be utilized parallel to the IMF information.
Comparing Chinas population growth of IMF data & the Function y= (1900)/1+2.42 (.97) x
Year Population in Millions
1983 1030.1
1992 1171.1
1997 1236.3
2000 1267. 4
2003 1292. 3
2005 1307. 6
2008 1327 .7
Population growth from equation y= (1900)/1+2.42 (.97)x Difference in population from original data to equation y= (1900)/1+2.42 (.97)x
1007.6
1135.5
1203.9
1243. 7
1282. 4
1307.5
1344. 1
-22.5
-35.6
-32.4
-23.7
-9.9
-.1
17
Comparing Chinas population growth of IMF data & the function P(t)=1946/ (1+2.61e-0.033t)
Year Population in Millions
1983 1030.1
1992 1171.1
1997 1236.3
2000 1267. 4
2003 1292. 3
2005 1307. 6
2008 1327 .7
Population growth from equation y= P(t)=1946/ (1+2.61e-0.033t) Difference in population from original data to equation
1036
1177.5
1252.7
1296. 3
1338. 4
1365.6
1405. 1
5.9
6.4
16.4
28.9
46.1
58
77.4
P(t)=1946/ (1+2.61e-0.033t)
Modified Equation for IMF Data 1. Equation for IMF data P(t)= K/ 1+Le-Mt P(t)=
(1915.2)/(1+2.64e -.033t)
I preferred this modification following inputting diverse numerical estimations into the unvarying and utmost of development. This modification varies from the IMF pose by 5 20 million individuals in by and large years, excluding the final few years where the distinctions are around 30 50 million. On the whole, this is a significant reliable pose that with enough modification might besides faultlessly fit the IMF information.
Year Populati on in millions
1983 1014.1
1992 1153.6
1997 1227.9
2000 1270.9
2003 1312.5
2005 1339.4
2008 1378.5
