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CHAPTER 3 PROJECT

CHAPTER 3 PROJECT World Population

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Law of Uninhibited Growth The Law of Uninhibited Growth states that, under certain conditions, the rate of change of a population is proportional to the size of the population at that time. One consequence of this law is that the time it takes for a population to double remains constant. For example, suppose a certain bacteria obeys the Law of Uninhibited Growth. Then if the bacteria take five hours to double from \(100\) organisms to \(200\) organisms, in the next five hours they will double again from \(200\) to \(400.\) We can model this mathematically using the formula \[ P(t) =P_{0}2^{t/D} \]

where \(P(t)\) is the population at time \(t,\) \(P_{0}\) is the population at time \(t=0,\) and \(D\) is the doubling time.

If we use this formula to model population growth, a few observations are in order. For example, the model is continuous, but actual population growth is discrete. That is, an actual population would change from \(100\) to \(101\) individuals in an instant, as opposed to a model that has a continuous flow from \(100\) to \(101.\) The model also produces fractional answers, whereas an actual population is counted in whole numbers. For large populations, however, the growth is continuous enough for the model to match real-world conditions, at least for a short time. In general, as growth continues, there are obstacles to growth at which point the model will fail to be accurate. Situations that follow the model of the Law of Uninhibited Growth vary from the introduction of invasive species into a new environment, to the spread of a deadly virus for which there is no immunization. Here, we investigate how accurately the model predicts world population.

The world population on January 1, 1959, was approximately \( 2.983435\times 10^{9}\) persons and had a doubling time of \(D=40\) years. Use these data and the Law of Uninhibited Growth to write a formula for the world population \(P=P( t)\). Use this model to solve Problems 2 through 4.
Find the rate of change of the world population \(P=P(t)\) with respect to time \(t\).
Find the rate of change on January 1, 2011 of the world population with respect to time. (Note that \(t=0\) is January 1, \(1959.)\) Round the answer to the nearest whole number.
Approximate the world population at the beginning of 2011. Round the answer to the nearest person.
According to the United Nations, the world population on January 1, 2009, was \(6.817727\times 10^{9}.\) Use \(t_{0}=2009,\) \(P_{0}=6.817727\times 10^{9}\), and \(D=40\) and find a new formula to model the world population \(P=P(t)\).
Use the new model from Problem 5 to find the rate of change of the world population at the beginning of 2011.
Compare the results from Problems 3 and 6. Interpret and explain any discrepancy between the two rates of change.
Use the new model to approximate the world population at the beginning of 2011. Round the answer to the nearest person.
The State of World Population report produced by the United Nations Population Fund indicates that the world population was estimated to reach 7 billion on October 31, 2011. Use the model from Problem 5 to approximate the world population on November 1, 2011. Compare the predicted population to the actual world population of 7 billion on October 31, 2011.
Use the original model (1959 data) to approximate the world population on November 1, 2011.
Discuss possible reasons for the discrepancies in the approximations of the 2011 population and the official number released by the UN. Was it more accurate to use the 1959 data or the 2009 data? Why do you think one set of data gives better results than the other?