question 9 of 10\none model of earths population growth is (p(t)=\frac{64}{1 + 11e^{-0.08t}}), where (t) is…

question 9 of 10\none model of earths population growth is (p(t)=\frac{64}{1 + 11e^{-0.08t}}), where (t) is measured in years since 1990, and (p) is measured in billions of people. which of the following statements are true?\ncheck all that apply.\na. in 1990, there were 5.33 billion people.\nb. the carrying - capacity of earth is 5.33 billion people.\nc. the population of earth will grow exponentially for a while but then start to slow down its growth.\nd. the population of earth is increasing by a steady rate of 8% per year.

question 9 of 10\none model of earths population growth is (p(t)=\frac{64}{1 + 11e^{-0.08t}}), where (t) is measured in years since 1990, and (p) is measured in billions of people. which of the following statements are true?\ncheck all that apply.\na. in 1990, there were 5.33 billion people.\nb. the carrying - capacity of earth is 5.33 billion people.\nc. the population of earth will grow exponentially for a while but then start to slow down its growth.\nd. the population of earth is increasing by a steady rate of 8% per year.

Answer

Explanation:

Step1: Analyze the population - growth model

The population - growth model is given by $P(t)=\frac{64}{1 + 11e^{-0.08t}}$, which is a logistic - growth model of the form $P(t)=\frac{L}{1+Ae^{-kt}}$, where $L$ is the carrying capacity, $A$ and $k$ are positive constants.

Step2: Find the population in 1990

When $t = 0$ (since $t$ is measured in years since 1990), we substitute $t = 0$ into the formula $P(t)=\frac{64}{1 + 11e^{-0.08t}}$. Then $P(0)=\frac{64}{1+11e^{0}}=\frac{64}{1 + 11}=\frac{64}{12}\approx5.33$ billion people. So statement A is true.

Step3: Determine the carrying capacity

For a logistic - growth model $P(t)=\frac{L}{1+Ae^{-kt}}$, the carrying capacity is $L$. In the given model $P(t)=\frac{64}{1 + 11e^{-0.08t}}$, the carrying capacity $L = 64$ billion people. So statement B is false.

Step4: Analyze the nature of growth

A logistic - growth model first shows exponential - like growth (when the population is small compared to the carrying capacity) and then slows down as the population approaches the carrying capacity. So statement C is true.

Step5: Analyze the growth rate

The given model is a logistic - growth model, not an exponential - growth model of the form $P(t)=P_0e^{rt}$. The growth rate in a logistic model is not a constant percentage rate. In an exponential growth model $P(t)=P_0e^{rt}$, the growth rate is $r$. But for a logistic model $P(t)=\frac{L}{1+Ae^{-kt}}$, the growth rate changes over time. So statement D is false.

Answer:

A. In 1990, there were 5.33 billion people. C. The population of Earth will grow exponentially for a while but then start to slow down its growth.