Sahana KanabarExtended Mathematics 10Miss Singhal4th May 2015

World Population

Population is a measure of all the inhabitants of a particular place. Earth now has over 7.2 billion people, and that number is growing. Recently the population has been increasing at a slower rate, however the growth now has more of a consequence on our daily lives.

Part 1

a)The formula for population growth is f(x) = P (1 + r)x. P = Original populationr = Rate of growth

In 1990, the world population reached 5.2 billion with an annual population growth rate of 1.75%. Using the formula f(x) = 5.2 (1.0175)x it is possible to make predictions about the future population. f(x) = 5.2 (1.0175)x

YearX-ValuePopulation (billions)

199005.200

199555.671

2005156.746

2010207.357

20506014.725

b)Using the same formula it is also possible to determine what the population was in previous years. As the population will be a decrease from the original population, the x value will be negative.

For example, 1950 was 40 years before 1990. Therefore the formula would be f(-40) = 5.2 (1.0175)-40f(-40) = 2.598aaaaaaaa

c)The population growth formula would also be useful for predicting milestones, such as when the population will double in size.10.4 = 5.2 (1.0175)x2 = 1.0175xaax = log1.0175 2= 39.95

After approximately 40 years, in 2030, the population will have doubled since 1990 to reach 10.4 billion. Based on the predictions for the year 2010 and 2050, this seems reasonable.

d)The one part of the formula that is changeable is the annual population growth rate. It is very important that the rate be as accurate as possible, as even a small change can create a large difference in the predictions. If the population grows at a faster rate, the r value will be changed.

Formula for a growth rate of 2%: f(x) = 5.2 (1.02)x

YearX-ValuePopulation (billions)

1.75%2%

199005.2005.200

199555.6715.741

2005156.7466.999

2010207.3577.727

20506014.72517.061

Part 2

1.Population growth does not increase at a constant rate, particularly over large periods of time. This model may not be accurate even after two years as the growth rate fluctuates. There are many factors that are not taken into account by the model, such as a boom in birth rates, an outbreak of a fatal disease, war, or other increase in accidental deaths. All the changes make it necessary that the model is re-evaluated regularly to ensure that the growth rate remains relevant and the predictions for the future are as accurate as possible.

2. A small change in the growth rate can have a large effect on the population over a large period of time. The predicted values for the first few consecutive years may be similar but over time the values will grow wider apart.

For example:YearX-ValuePopulation (billions)

1.70%1.75%1.80%

199005.2005.2005.200

199555.6575.6715.685

2005156.6966.7466.795

2010207.2857.3577.429

20506014.29814.72515.166

YearX-ValuePercentage Error (%)

1.70%1.75%1.80%

199000.0000.0000.000

199550.5300.2820.035

2005153.4954.2104.901

2010206.1085.7467.928

205060???

As the table illustrates, after 5 years the estimated population differentiates by 28 million, a small error margin considering the initial difficulty in determining the exact population. After 15 years the estimated populations have a difference of 100 million. After 60 years, the difference is almost 868 million. This could have many complications regarding resource allocation, area development, and many other things that are required to prepare for population increase. However, the margin of error leaves some room for the unpredictable changes in population that are constantly changing the growth rate. It takes time for the population to grow, meaning that the growth rate will only experience small changes. Even so, a slight change in the growth rate used in the model will have a major impact on any future population predictions.

3. In order to determine the accuracy of the models, the predicted values need to be compared to the actual population. Below 2010 is used as an example with the growth rate of 1.75%, 2%, and the actual population.

YearX-ValuePopulation (billions)

1.75%2%Actual

2010207.3577.7276.840

YearX-ValuePercentage Error (%)

1.75%2%

2010207.037 11.479

The prediction made for the 2010 population using both models were considerably higher than the actual population. In 1990 the growth rate of the population was 1.75% but over time the growth rate slowed down, taking the world population longer to reach 7 billion. A 2% population growth rate is even higher than 1.75%, which gave it a larger percentage error. As time goes on, the percentage error will increase, making the growth rate obsolete.

4. Both models used assume a constant rate of population growth each year. The predictions may be reasonable for the first few years following the creation of the model but after a significant period of time they become unreliable due to the changes in growth rate. On average, the official predictions made by the United Nations and the World Bank are off by about 6%. A more accurate model could be made using the average growth rates of the population from 1990 to 2010 but the models used the growth rate from 1990 and before. For a prediction 20 years in the future using these models is not reasonable.

5. The current population growth rate is around 1.14%, a significant decrease from the given growth rate of 1.75%.

YearX-ValuePopulation (billions)

1.75%1.14%

199005.2005.200

199555.6715.574

2005156.7466.406

2010207.3576.867

20506014.72511.975

While the population may have been growing at a rate of 1.75% in 1990, by 2010 the growth rate of 1.14% gives much more accurate predictions. While it could increase preparedness for growth to use predictions that overestimate population, it could have a big impact on actual numbers and make planning for population increase unreliable. Although the growth rate may have slowed down for many reasons, due to the increasing size of the actual population this is still a significant population increase every year. Because it is an exponential equation, the predicted population is growing from a percentage of the total population.

6. It is not reasonable to assume a constant annual rate of growth for the world population. Population growth rates are affected by birth and death rates, which are constantly changing. Unless one person was born for every person that died, the population is either increasing or decreasing. The average population growth rate can be applied but will only be accurate for a short span of time. Growth rates fluctuate and cannot be modeled as a constant if completely accurate answers are wanted. For example, the population growth rate was 1.75% in 1990, but was 1.5% in 1995 and is now around 1.14%. Even two consecutive years can have a large difference. The world population growth rate was 1.7% in 1962 and jumped to 2.1% in 1962. Specific countries can also have their population affected by migration and emigration.

7. The more specific the results are, the less accurate they also are. Population is measured in millions or billions so as to give an idea of what the population is without getting into the details. Predictions are made on an assumed growth rate that may or may not hold true for the time frame under which the growth is predicted. The small numbers are constantly changing as people are born or die, but it takes time for those numbers to equal millions. Knowing the population in millions is still useful for showing change and determining the general impact the population may have.

8. Today China has a population of approximately 1.400 billion (2015), the largest population for any country in the world. In order to slow the population growth, the government introduced a one-child policy in late 1979 that greatly reduced the countrys birth rate. Before the policy people were encouraged to have many children to increase the countrys workforce, but it soon became clear that with the 1.9% rate of population growth accommodating this population would quickly become unsustainable. Benefits to families with only one child included increased access to education, childcare, and healthcare. The policy has been very successful, as the current population growth rate is 0.7%. The government sought a solution to a future problem by implementing the one-child policy as a precaution against a potential lack of resources. This policy has created some new problems, such as an aging population and an increase in the gender gap due to cultural values, but was successful at addressing the population growth.

9.In a real life context, it is very important to be able to model population growth. The population has direct implications for political stability, resource (food, water, shelter, jobs) security, and climate change. The population growth rate will keep changing; however making predictions about the future population allows us to be more prepared for the effects it will have. The initial increase in population that happened after the end of World War II has already put a strain on the worlds resources. After the Great Depression and a long period of living thriftily made way for the baby boomers time of prosperity. They created large debt for their countries and founded the companies that are destroying our environment. As a generation they had little regard for the future and the lifestyle of excess has left many problems we are scrambling to fix. Monitoring the population, and its future predictions allows us to be more prepared for any changes, and prevent many problems.

10. If the worlds population increased much more rapidly than expected there would be a lot of problems. A finite amount of land and resources is incapable of supporting a potentially limitless increasing population. Overpopulation is already a problem in many countries, leading to a lack of resources, particularly with the unsustainable lifestyles we already lead. Due to the finite nature of many of our resources, including food and water, major conflicts would arise around securing the resources and allocating them to people. World population affects political discussions on sustainability and development. Even with the increased need for a sustainable lifestyle, we are not equipped to handle a rapid increase in population.