Questions: Assume the carrying capacity of the Earth is 16 billion. Use the 1960 annual growth rate of 2.1% and population of 3 billion to predict the base growth rate and current growth rate with a logistic model. Assume a current world population of 7.8 billion. How does the predicted growth rate compare to the actual growth rate of about 1.1% per year? What is the base growth rate? 2.5846% (Round to four decimal places as needed.) What is the estimated current growth rate? 1.32% (Round to two decimal places as needed.) How does the estimated growth rate compare to the actual current growth rate? The predicted growth rate is the actual growth rate. - smaller than - the same as - larger than

Solution Steps

To solve this problem, we will use the logistic growth model, which is defined by the equation:

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

where:

• \( P(t) \) is the population at time \( t \),
• \( K \) is the carrying capacity,
• \( P_0 \) is the initial population,
• \( r \) is the base growth rate,
• \( t \) is the time in years.

1. Base Growth Rate Calculation: Use the given initial conditions (1960 population and growth rate) to solve for the base growth rate \( r \).
2. Current Growth Rate Calculation: Use the logistic model to estimate the current growth rate given the current population.
3. Comparison: Compare the estimated current growth rate to the actual growth rate.

To find the base growth rate \( r \), we use the initial growth rate and the carrying capacity. The calculated base growth rate is given by:

\[ r_{\text{base}} = 0.0170625 \]

Expressed as a percentage, this is:

\[ r_{\text{base}} \times 100 = 1.7063\% \]

Next, we estimate the current growth rate \( r_{\text{current}} \) using the logistic model. The calculated estimated current growth rate is:

\[ r_{\text{estimated current}} = 0.00874453125 \]

Expressed as a percentage, this is:

\[ r_{\text{estimated current}} \times 100 = 0.8745\% \]

We compare the estimated current growth rate to the actual growth rate of \( 0.011 \) (or \( 1.1\% \)). The comparison shows that:

\[ 0.00874453125 < 0.011 \]

Thus, the estimated growth rate is smaller than the actual growth rate.

Final Answer