To find the population of llamas after 10 years, given that the population is decreasing by 10 percent per year, we can use the exponential decay equation:
\[ P(t) = P_0 \cdot (1 - r)^t \]
where:
- \( P(t) \) is the population after time \( t \),
- \( P_0 \) is the initial population,
- \( r \) is the rate of decrease (as a decimal),
- \( t \) is the time in years.
Given:
- \( P_0 = 50,000 \)
- \( r = 0.10 \)
- \( t = 10 \)
We plug these values into the equation:
\[ P(10) = 50,000 \cdot (1 - 0.10)^{10} \] \[ P(10) = 50,000 \cdot (0.90)^{10} \]
Now, we calculate \( (0.90)^{10} \):
\[ (0.90)^{10} \approx 0.3487 \]
Now we calculate \( P(10) \):
\[ P(10) = 50,000 \cdot 0.3487 \approx 17,435 \]
Rounding gives us approximately 17,433.
Therefore, the population after 10 years is approximately 17,433.