MATH: Population Extrapolation Using a Mathematical Model Project

This project presents a mathematical model for extrapolating population in a given area, addressing the challenges of inaccurate population predictions in socio-economic and demographic planning. The model, accompanied by Newton’s divided difference formula, aims to provide accurate population figures, crucial for resource management and economic planning. The report discusses the model's derivation, application, and error analysis, highlighting its potential for improving census accuracy and aiding countries in effectively planning for population growth. It references existing literature on population dynamics and logistic models, emphasizing the Newton’s divided difference formula for a comprehensive result of the population. The project concludes that the model offers a uniform criterion for determining population growth at different time intervals, essential for better economic resource planning and development.

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Extrapolation of the population using the mathematical model
Abstract
This project is a representation of the extrapolation of the population in a given area
using the mathematical model. This model can be used at any time to find the population of the
area. A mathematical software can also be used in determining the parameters after the
derivation of the model since it was difficult to find the solution (Alpers, 2015). This model is
accompanied by the Newton’s divided difference formula. The model experienced just about
10% error when it was applied. Therefore this model is recommendable and should be applied by
most of the countries in the world during the census to help them in the extrapolation of the
population (Izard, 2011).
Introduction
There have been difficulties in the socio-economic and demographic planning in most of
the countries in the world due to the census that is done in this countries. This difficulty has been
created as a result of the wrong prediction of the size of the population in a country. This
mathematical model should be used studies and implemented by countries to help in the
determination of most accurate population figures in a country (Loch, 2007).
The greatest challenge that world is facing today is the excess growth in population if
compared to the development rate of the natural resources (Gillis, 2012). Since the socio-
economic development and demographic improvement depends on the population growth in a
country is therefore important for the countries to adopt the implementation of this formula
counteract with the growth rate. The determination in the population growth helps marketing in
business, research and in the planning of the economic and financial growth nationally (Bolt,
2012).

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Literature review
When determining the population growth there are Mathematical model dynamics of the
population in a country and growth logistic model. The theorem of the dynamics of population
states that there is exponential growth in population and a linear growth in the supply of food.
This theorem has created a belief that increase in the food leads to increase in the population
(Gardner, 2015).
However, the growth logistic model theorem states that growth of population in an area
depends on both the size of the population and the carrying capacity effects that might cause
growth limitation. The two methods above does not give a comprehensive result on the rate of
growth and proper exponential growth in this area of target hence the Newton’s divided
difference formula must be applied to provide a complete and comprehensive result of the
population of a country(Loch,2011).
Discussion
The division of the successive difference can be given by;
The population growth depends on the unit of the population hence it is expressed by

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Where the value of P is then population and
Indicates the population chance with respect to time (Garcia, 2006)
Assuming the population of the of a given area is as shown in the graph bellow
The quantity bellow obtained from the graph;
Where d and q are constants. This equation is gives the linearity between the population and the
divided difference. Thus;
When both sides of the equation ids integrated, then we get the equation below.

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Getting the partial fraction;

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Comparing this to the population of a country where the x=0 and unit time=1 year G=30,
y=6.65.From the equation above we get

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Hence d is given by
Thus the final equation is;
Conclusion
It is important to have a proper model in the determination of the population in a country.
This model helps in the improvement of the economic resources by proving a good planning.
The formula Newton’s divided difference has provided uniform criteria for determining the
growth of the population in a country. This is, therefore, the best way to determine the
population at different time intervals (Boaler, 2012).

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