The rate of growth of a population is proportional to the population. If the population in 2000 was 3 million and is 2020, the population was 3.21 million, what is the predicted population for the year 2030?

This problem deals with exponential population growth, where the rate of growth is proportional to the current population. We can model this with the equation:

\[
P(t) = P_0 e^{kt}
\]

Where:
- \( P(t) \) is the population at time \( t \).
- \( P_0 \) is the initial population (at \( t = 0 \)).
- \( k \) is the growth rate constant.
- \( t \) is the time in years since the initial measurement.

### Step 1: Set up the initial conditions.
- The initial population in the year 2000 is 3 million, so \( P_0 = 3 \).
- The population in the year 2020 is 3.21 million. This is 20 years after 2000, so at \( t = 20 \), \( P(20) = 3.21 \).

### Step 2: Find the growth constant \( k \).
From the equation \( P(t) = P_0 e^{kt} \), we can express the population at 2020 as:
\[
P(20) = 3 e^{20k} = 3.21
\]

Solve for \( k \):
\[
e^{20k} = \frac{3.21}{3} = 1.07
\]
Taking the natural logarithm of both sides:
\[
20k = \ln(1.07)
\]
\[
k = \frac{\ln(1.07)}{20}
\]

### Step 3: Calculate the predicted population for 2030.
Now, to find the population in 2030 (\( t = 30 \) years after 2000), we use the equation:
\[
P(30) = 3 e^{30k}
\]

Let's compute the values.The predicted population for the year 2030 is approximately **3.32 million**.

Would you like more details or have any questions? 

Here are 5 related questions to expand your understanding:
1. How does the growth constant \( k \) affect the population prediction?
2. Can this model be applied to other growth phenomena like bacteria or investments?
3. What assumptions are made in this population growth model?
4. How would the population be affected if the growth rate changes over time?
5. How can we modify the model to account for factors like migration or policy changes?

**Tip:** Exponential growth models are useful for predicting population changes, but they work best when the growth rate remains consistent over time.

The rate of growth of a population is proportional to the population. If the population in 2000 was 3 million and in 2020, the population was 3.21 million, what is the predicted population for the year 2030?

This problem explores exponential population growth, where the growth rate is proportional to the current population. Given that the population was 3 million in 2000 and 3.21 million in 2020, learn how to predict the population for 2030 using exponential growth modeling. Suitable for high school students in Grades 10-12.

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