To estimate the growth rate $r$ using the given exponential growth model, we start by using the equation for population growth:

$P(t) = P(0) e^{rt}$

Where:

• $P(t)$ is the population at time $t$,
• $P(0)$ is the initial population at $t = 0$,
• $r$ is the growth rate,
• $t$ is the time in years since 2000.

Step 1: Identify the Known Values

• $P(0) = 281$ million (population in the year 2000),
• $P(10) = 310$ million (population in the year 2010),
• $t = 10$ years (since 2010 is 10 years after 2000).

Step 2: Apply the Exponential Growth Equation

We have:

$310 = 281 \times e^{10r}$

Step 3: Solve for the Growth Rate $r$

First, isolate the exponential term:

$\frac{310}{281} = e^{10r}$

Next, take the natural logarithm on both sides:

$\ln\left(\frac{310}{281}\right) = 10r$

Finally, solve for $r$:

$r = \frac{1}{10} \ln\left(\frac{310}{281}\right)$

Let's calculate this value.

The estimated growth rate $r$ is approximately $0.00982$ when rounded to five decimal places.

Would you like any further details on this calculation, or do you have any other questions?

Here are 5 related questions to consider:

1. How would the population change if the growth rate doubled?
2. What would be the population in 2020 using the exponential growth model?
3. How does exponential growth differ from logistic growth?
4. What are some factors that could cause the growth rate to change over time?
5. How does the carrying capacity impact the logistic growth model?

Tip: Always check the units when solving exponential growth problems, as they can affect the interpretation of your results.