Question

The llama population on an island is decreasing by 10
percent per year. Use an exponential equation find
the population after 10 years if the population
started with 50,000 llamas at the end of year 1.

Answer 1

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**.