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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. $$ \begin{array}{cc} {x, \text { Number of Years }} & {y, \text { World Population }} \\ {\text { after } 1949} & {\text { (billions) }} \\ {1(1950)} & {2.6} \\ {11(1960)} & {3.0} \\ {21(1970)} & {3.7} \\ {21(1970)} & {4.5} \\ {41(1990)} & {5.3} \\ {51(2000)} & {6.1} \\ {61(2010)} & {6.9} \end{array} $$ 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 $$ f(x)=\frac{12.57}{1+4.11 e^{-0.026 x}} $$ 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

01

Understand the Function

In a logistic function, the limit is the so-called 'carrying capacity', which represents the maximum number of individuals that the environment can support indefinitely. In this function, it is given by the numerator.

02

Identify Carrying Capacity

In our logistic model, the carrying capacity is the numerator of the fraction, or 12.57. This is the maximum population that the model says the Earth can sustain, in billions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

World Population

World population refers to the total number of humans currently inhabiting the Earth. This number fluctuates over time due to various factors such as birth rates, death rates, and migration.
It’s essential to understand these dynamics because they help us predict future trends and make decisions based on this data. Historically, the world population has seen periods of rapid growth, especially during certain decades due to technological and medical advancements, which have reduced mortality rates.
Understanding how the population might change can aid in planning for resources, infrastructure and can help governments and organizations to strategize for sustainable development. By using mathematical models, like the logistic growth model, we can estimate future changes and challenges tied to population growth.

Carrying Capacity

Carrying capacity is a key concept in ecological studies and refers to the maximum number of individuals that an environment can sustainably support. In the context of world population, it involves the limits to which Earth can provide resources such as food, water, and shelter for humans.
In a logistic growth model, carrying capacity is the horizontal asymptote. This represents the point where the growth of the population slows and stabilizes because resources become limited.
The carrying capacity varies due to changes in technology, policies, and social structures. For our logistic model of world population, the carrying capacity was identified as 12.57 billion. This suggests that Earth can sustain up to 12.57 billion people under the assumptions of this particular model.

Logistic Regression

Logistic regression is a statistical method used to model the probability of a certain class or event. It’s useful in cases where growth initially follows exponential patterns but eventually tapers off, such as world population growth.
This technique was used to fit a model to world population data by creating a function that can predict future population sizes based on past trends in the data.

• The function obtained from logistic regression is shaped like an "S." It grows rapidly at first but then levels off as it approaches carrying capacity.
• Using this method, statisticians can predict when changes in population may occur, such as when growth might slow or reverse.

Logistic regression helps us understand complex relationships and trends in demographic data, providing a refined approach for making longer-term predictions.

Exponential Growth

Exponential growth is a pattern of data that shows greater increases over time, creating a J-shaped curve. It occurs when the growth rate of the value of a mathematical function is proportional to the function's current value, leading to continuous and rapid increases.
In the context of population growth, exponential growth happens when the number of people increases rapidly over a short period, due to high birth rates and falling death rates.
However, this type of growth is not sustainable indefinitely because resources eventually become limited. That's where the logistic growth model comes into play, providing a more realistic prediction.

• Exponential growth initially drives human populations upwards but isn't realistic over the long term due to environmental limits.

Understanding this concept is vital for predicting shifts in population dynamics and making sustainable development choices.