One model of Earth's population growth is [tex]$P(t)=\frac{64}{\left(1+11 e^{-.08 t}\right)}$[/tex] measured in years since 1990, and P is measured in billions of people. Which of the following statements are true?
Check all that apply.
A. The population of Earth is increasing by a steady rate of [tex]$8 \%$[/tex] per year.
B. The population of Earth will grow exponentially for a while but then start to slow down its growth.
C. In 1990, there were 5.33 billion people.
D. The carrying capacity of Earth is 5.33 billion people.
Answer (2)
Calculate the population in 1990: P ( 0 ) = 1 + 11 64 = 3 16 ≈ 5.33 billion.
Determine that the population growth slows down as it approaches the carrying capacity of 64 billion, indicating logistic growth.
Verify that the population in 1990 is approximately 5.33 billion.
Conclude that statements B and C are true: B and C .
Explanation
Problem Analysis We are given a population growth model P ( t ) = ( 1 + 11 e − .08 t ) 64 , where t is the number of years since 1990 and P is the population in billions. We need to determine which of the given statements are true.
Analyzing Statement A Statement A says the population increases by a steady rate of 8% per year. To check this, we can calculate the population in 1990 and 1991 and see if the increase is 8%. The population in 1990 is P ( 0 ) = 1 + 11 e − 0.08 ( 0 ) 64 = 1 + 11 64 = 12 64 = 3 16 ≈ 5.33 billion. The population in 1991 is P ( 1 ) = 1 + 11 e − 0.08 ( 1 ) 64 ≈ 1 + 11 ( 0.923 ) 64 ≈ 1 + 10.153 64 ≈ 11.153 64 ≈ 5.738 billion. The percentage increase is 5.33 5.738 − 5.33 × 100 ≈ 5.33 0.408 × 100 ≈ 7.65% . Since the percentage increase is not exactly 8%, statement A is false.
Analyzing Statement B Statement B says the population grows exponentially for a while but then slows down. This is characteristic of logistic growth models. As t increases, e − 0.08 t approaches 0, so P ( t ) approaches 1 + 0 64 = 64 . This means the population growth slows down as it approaches the carrying capacity of 64 billion. So, statement B is true.
Analyzing Statement C Statement C says in 1990, there were 5.33 billion people. We already calculated P ( 0 ) = 3 16 ≈ 5.33 billion. So, statement C is true.
Analyzing Statement D Statement D says the carrying capacity of Earth is 5.33 billion people. The carrying capacity is the limit of P ( t ) as t approaches infinity, which we found to be 64 billion, not 5.33 billion. So, statement D is false.
Conclusion Therefore, the true statements are B and C.
Examples
Logistic growth models, like the one in this problem, are used in various fields such as ecology, economics, and epidemiology. For example, in epidemiology, these models can help predict the spread of a disease within a population, considering factors like transmission rates and recovery rates. Understanding these models allows public health officials to implement effective strategies to control and mitigate the impact of infectious diseases, optimizing resource allocation and minimizing the overall burden on society. Similarly, in economics, logistic models can be used to describe the adoption rate of a new technology or product in a market, helping businesses make informed decisions about production, marketing, and investment.
The true statements about Earth's population growth model are B (the population grows exponentially for a while but then slows down) and C (in 1990, there were approximately 5.33 billion people). Statement A is false since the growth rate is not a steady 8%, and Statement D is false as the carrying capacity is actually 64 billion. Thus, the correct answer is B and C.
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