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We are given the exponential model \( A = 481.5 e^{0.01t} \), where \( A \) is the population of a country in millions, and \( t \) is the number of years after 2003. The goal is to determine the year when the population will be 576 million.

We can follow these steps to solve the problem:

1. Set \( A = 576 \) in the given model:
   \[
   576 = 481.5 e^{0.01t}
   \]

2. Solve for \( t \). First, divide both sides by 481.5:
   \[
   \frac{576}{481.5} = e^{0.01t}
   \]
   Simplifying:
   \[
   1.196 = e^{0.01t}
   \]

3. Take the natural logarithm (ln) of both sides:
   \[
   \ln(1.196) = 0.01t
   \]
   Using a calculator for the natural logarithm of 1.196:
   \[
   0.178 = 0.01t
   \]

4. Solve for \( t \):
   \[
   t = \frac{0.178}{0.01} = 17.8
   \]

Thus, \( t \approx 18 \) years. Since \( t \) represents the number of years after 2003, we add 18 to 2003:

\[
2003 + 18 = 2021
\]

Therefore, the population of the country will be 576 million in the year **2021**.

Would you like further details or explanations?

Here are 5 related questions for deeper understanding:
1. What is the significance of the constant 481.5 in the model?
2. How does the base \( e \) affect the population growth over time?
3. What happens to the population if the growth rate (0.01) changes to a higher value?
4. How can you adjust the formula if you are calculating years from a different starting point?
5. How would you use this model to predict the population in future years beyond 576 million?

**Tip:** When solving exponential growth models, always remember to take the natural logarithm to handle equations involving the exponential function \( e^x \).

The exponential model A = 481.5e^(0.01t) describes the population, A, of a country in millions, t years after 2003. Use the model to determine when the population of the country will be 576 million.

Learn how to solve an exponential growth problem using the model A = 481.5e^(0.01t). This example demonstrates how to determine the year when a country's population reaches 576 million, involving key concepts such as natural logarithms and growth models. Suitable for high school students in Grades 10-12.

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