Let's work through parts (c) and (d) of the problem step by step.

Part (c)

Problem Statement: The task is to estimate the population of the United States in 1970 using the logistic model. Then, you need to compare this estimate to the actual population of 203 million in 1970. The given logistic model is:

$P(t) = \frac{1}{1 + \frac{1}{K} \cdot \left( \frac{P_0}{K - P_0} \right) \cdot e^{-rt}}$

where:

• $P(t)$ is the population at year $t$,
• $P_0$ is the initial population (at $t = 0$),
• $K$ is the carrying capacity (maximum population the environment can sustain),
• $r$ is the growth rate.

To apply this, you'll first need to estimate the parameters $K$, $r$, and $P_0$.

Steps to solve:

1. Determine the values for $K$, $r$, and $P_0$.

   • From the table, choose a year to represent $t = 0$. Typically, you might choose 1790 (first year in the table).
   • $P_0$ would then be the population in 1790, which is $P_0 = 3.929$ million.
   • $K$ needs to be estimated from the data. Since the U.S. population was rapidly increasing, you'll need to assume a plausible $K$ value or calculate one based on observed growth.

2. Estimate $K$ and $r$ by fitting the logistic model to the given data. This typically requires more advanced statistical or computational tools.

3. Predict $P(t)$ for $t = 180$ years after 1790 (i.e., in 1970).

4. Compare the predicted value $P(180)$ to the actual population in 1970.

Part (d)

Problem Statement: The task is to determine when the population in the U.S. was growing most rapidly and to explain how the logistic model determines this. The population grows most rapidly at the inflection point, where the second derivative of $P(t)$ with respect to ( t \