Question

question 10 of 10
one 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?
check all that apply.
a. the population of earth is increasing by a steady rate of 8% per year.
b. the population of earth will grow exponentially without bound.
c. the carrying capacity of earth is 64 billion people.
d. in 1990, there were 5.33 billion people.

Explanation:

Step1: Analyze option A

The given function $P(t)=\frac{64}{1 + 11e^{-0.08t}}$ is a logistic - growth function, not a simple exponential growth function of the form $y = a(1 + r)^t$. So the population is not increasing at a steady rate of 8% per year. Option A is false.

Step2: Analyze option B

Logistic - growth functions have a carrying capacity and do not grow exponentially without bound. As $t
ightarrow\infty$, $e^{-0.08t}
ightarrow0$, and $P(t)
ightarrow64$. Option B is false.

Step3: Analyze option C

For a logistic - growth function of the form $P(t)=\frac{L}{1 + Ae^{-kt}}$, $L$ is the carrying capacity. In the function $P(t)=\frac{64}{1 + 11e^{-0.08t}}$, $L = 64$. So the carrying capacity of Earth is 64 billion people. Option C is true.

Step4: Analyze option D

When $t = 0$ (corresponding to the year 1990), $P(0)=\frac{64}{1+11e^{0}}=\frac{64}{1 + 11}=\frac{64}{12}\approx5.33$ billion people. Option D is true.

Answer:

C. The carrying capacity of Earth is 64 billion people., D. In 1990, there were 5.33 billion people.