Problem 42 - College Algebra

Question

Shown, again, in the following table is world population, in billions, for seven selected years from 1950 through \(2010 .\) Using a graphing utility's logistic regression option, we obtain the equation shown on the screen. (TABLE CANNOT COPY) We see from the calculator screen at the bottom of the previous page that a logistic growth model for world population, \(f(x),\) in billions, \(x\) years after 1949 is $$f(x)=\frac{12.57}{1+4.11 e^{-0.52 h x}}$$ Use this function to solve Exercises \(38-42\) According to the model, what is the limiting size of the population that Earth will eventually sustain?

Step-by-Step Solution

Answer

The estimated maximum size of the population that Earth can eventually support, according to the given model, is 12.57 billion.

Step 1: Apply logarithm properties

Use the product rule \(\log(ab) = \log a + \log b\), quotient rule \(\log(a/b) = \log a - \log b\), and power rule \(\log(a^n) = n\log a\).

Step 2: Simplify

The result is The estimated maximum size of the population that Earth can eventually support, .

Key Concepts

Carrying Capacity

The concept of 'carrying capacity' is pivotal in understanding the limitations of our environment. It refers to the maximum number of individuals, whether people, animals, or plants, that a particular environment can support sustainably over time without degradation. In the context of the world population, carrying capacity relates to the maximum human population that the Earth can support given the available resources, such as food, water, and habitable land.

The logistic growth model presents carrying capacity as an upper boundary that populations tend to approach but never exceed. This balance occurs as the population growth rate slows down as resource limits are reached. Considering the provided logistic growth model, \(f(x)=\frac{12.57}{1+4.11 e^{-0.52 h x}}\), the carrying capacity is suggested to be 12.57 billion. This means the model predicts that the world population will approach but not surpass this number under the current model assumptions.

World Population

The study of world population dynamics is an area of major interest for demographers, economists, and environmental scientists. The world population has been growing at unprecedented rates since the industrial revolution due to advancements in medicine, agriculture, and technology. On one hand, this growth represents a triumph of human innovation; on the other hand, it raises concerns about sustainability and living standards.

By utilizing models like the logistic growth equation, scientists attempt to forecast future population trends. The provided logistic model used historical data and projected that the world's population will stabilize around 12.57 billion because of natural limitations and resource scarcity implying a carrying capacity. This projection contributes to policy planning, resource allocation, and understanding potential environmental impacts.

Logistic Regression

Logistic regression is not to be confused with the logistic growth model, although they are related. It is a statistical method used for binary classification that can also be adapted to model the rate of occurrence of events having probabilities that can change over time. In the context of world population growth, logistic regression is the process by which we analyze past population data to determine the parameters of a logistic growth model.

Through logistic regression analysis, we can derive functions like \(f(x)=\frac{12.57}{1+4.11 e^{-0.52 h x}}\). This function represents the model that is used to estimate the growth of the world population as a function of time, with predictions subject to the carrying capacity and growth rates inferred from the regression.