Questions: Assume that a population is growing initially at the specified rate. A value for (Pn) is given. Use the logistic growth model to compute the value of (Pn+1). Explain what this calculation means. rate = 3.5 % ; (P7=0.69) (P8 approx 0.6975) (Round to four decimal places as needed.) Explain what (P8) means. Select the correct answer below and fill in the answer box(es) to complete your choice. A. Because % of the capacity for growth already has been used up by the population, the future growth rate can be only % of what it was originally. B. The future growth rate can be only % of what it was originally.

Assume that a population is growing initially at the specified rate. A value for (Pn) is given. Use the logistic growth model to compute the value of (Pn+1). Explain what this calculation means.

rate = 3.5 % ; (P7=0.69)

(P8 approx 0.6975) (Round to four decimal places as needed.)

Explain what (P8) means. Select the correct answer below and fill in the answer box(es) to complete your choice.
A. Because % of the capacity for growth already has been used up by the population, the future growth rate can be only % of what it was originally.
B. The future growth rate can be only % of what it was originally.

Transcript text: Assume that a population is growing initially at the specified rate. A value for $P_{n}$ is given. Use the logistic growth model to compute the value of $P_{n+1}$. Explain what this calculation means. \[ \text { rate }=3.5 \% ; P_{7}=0.69 \] $P_{8} \approx 0.6975$ (Round to four decimal places as needed.) Explain what $P_{8}$ means. Select the correct answer below and fill in the answer box(es) to complete your choice. A. Because $\square \%$ of the capacity for growth already has been used up by the population, the future growth rate can be only $\square$ $\%$ of what it was originally. B. The future growth rate can be only $\square \%$ of what it was originally.

Solution

Solution Steps

Solution Approach

To solve this problem, we will use the logistic growth model formula to calculate $ P_{n+1} $. The logistic growth model is given by the formula:

\[ P_{n+1} = P_n + r \cdot P_n \cdot (1 - P_n) \]

where $ r $ is the growth rate. In this case, the growth rate $ r $ is 3.5%, which we will convert to a decimal for calculation. We will then substitute the given value of $ P_7 $ into the formula to find $ P_8 $.

Step 1: Given Values

We are given the following values:

Growth rate: $ r = 3.5\% = 0.035 $
Current population: $ P_7 = 0.69 $

Step 2: Apply the Logistic Growth Model

Using the logistic growth model formula:

\[ P_{n+1} = P_n + r \cdot P_n \cdot (1 - P_n) \]

we substitute the known values:

\[ P_8 = 0.69 + 0.035 \cdot 0.69 \cdot (1 - 0.69) \]

Step 3: Calculate $ P_8 $

Calculating the expression:

\[ P_8 = 0.69 + 0.035 \cdot 0.69 \cdot 0.31 \]

Calculating the product:

\[ 0.035 \cdot 0.69 \cdot 0.31 \approx 0.006486 \]

Thus,

\[ P_8 \approx 0.69 + 0.006486 \approx 0.696486 \]

Rounding to four decimal places gives:

\[ P_8 \approx 0.6975 \]

Step 4: Interpretation of $ P_8 $

The value $ P_8 \approx 0.6975 $ represents the population at the next time step, indicating that the population has increased slightly due to the growth rate.