To model the declining llama population using an exponential equation, we can use the formula for exponential decay:
\[ P(t) = P_0 \cdot (1 - r)^t \]
where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( r \) is the rate of decrease (as a decimal),
- \( t \) is the time in years.
Given:
- Initial population (\( P_0 \)) = 50,000 llamas
- Decrease rate (\( r \)) = 10% = 0.10
- Time (\( t \)) = 10 years
Now, we can plug these values into the formula:
\[ P(10) = 50000 \cdot (1 - 0.10)^{10} \]
Calculating:
\[ P(10) = 50000 \cdot (0.90)^{10} \]
Now, we calculate \( (0.90)^{10} \):
\[ (0.90)^{10} \approx 0.3487 \]
Thus,
\[ P(10) \approx 50000 \cdot 0.3487 \approx 17435 \]
Therefore, the population after 10 years would be approximately 17,435 llamas.
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