The population of a country dropped from 52.5 million in 1995 to 44.9 million in 2008. Assume that P(t), the population, in millions, t years after 1995, is decreasing according to the exponential decay model. After how many years will the population of the country be 2 million, according to this model?

To solve this problem, let's start by using the exponential decay model formula:

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

where:
- \( P(t) \) is the population at time \( t \) (in years after 1995),
- \( P_0 \) is the initial population (at \( t = 0 \)),
- \( k \) is the decay constant,
- \( t \) is the time in years after 1995.

### Step 1: Determine the Initial Population (\( P_0 \))
From the problem, we know:
- In 1995 (when \( t = 0 \)), the population was \( P_0 = 52.5 \) million.

### Step 2: Use Information to Find \( k \)
We are given that in 2008 (which is 13 years after 1995, so \( t = 13 \)), the population was \( P(13) = 44.9 \) million. We can plug this into the formula:

\[
44.9 = 52.5 e^{-13k}
\]

Solving for \( k \):

1. Divide both sides by 52.5:
   \[
   \frac{44.9}{52.5} = e^{-13k}
   \]

2. Simplify:
   \[
   0.8552 = e^{-13k}
   \]

3. Take the natural logarithm (ln) of both sides:
   \[
   \ln(0.8552) = -13k
   \]

4. Solve for \( k \):
   \[
   k = -\frac{\ln(0.8552)}{13}
   \]

   Let's calculate \( k \).

\[
k \approx 0.0121
\]

### Step 3: Find the Time \( t \) When the Population is 2 Million
We now want to find \( t \) when \( P(t) = 2 \) million. Using the same model equation:

\[
2 = 52.5 e^{-0.0121t}
\]

1. Divide both sides by 52.5:
   \[
   \frac{2}{52.5} = e^{-0.0121t}
   \]

2. Simplify:
   \[
   0.0381 = e^{-0.0121t}
   \]

3. Take the natural logarithm of both sides:
   \[
   \ln(0.0381) = -0.0121t
   \]

4. Solve for \( t \):
   \[
   t = -\frac{\ln(0.0381)}{0.0121}
   \]

   Let's calculate \( t \).

\[
t \approx 318.55
\]

### Conclusion
The population of the country will reach 2 million approximately **319 years** after 1995, or around the year **2314**.

Would you like a step-by-step explanation of any part of this process, or do you have any questions?

Here are 5 related questions to consider:

1. How would the model change if the population were increasing instead of decreasing?
2. What factors in real life might cause the actual population decline to differ from this model's predictions?
3. How can we modify this model to account for a constant rate of population decline rather than an exponential one?
4. What is the half-life of the population decline based on the decay constant \( k \)?
5. How would the model change if the decay constant \( k \) were not constant over time?

**Tip:** When dealing with exponential models, always check if the assumptions (like constant rate \( k \)) make sense in the real-world context.

Explore how to calculate the time it takes for a population to decline to 2 million, given its drop from 52.5 million in 1995 to 44.9 million in 2008. This problem uses the exponential decay model and logarithmic equations, suitable for students in Grades 10-12.

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