Question

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**.
make this like a 8th grader wrote it

Answer 1

To figure out how many llamas are left after a certain number of years when their population is slowly decreasing, we can use a formula called exponential decay. The formula looks like this:

\[
P(t) = P_0 \cdot (1 - r)^t
\]

Here’s what the letters mean:
- \( P(t) \) is the llama population after time \( t \),
- \( P_0 \) is the starting number of llamas,
- \( r \) is how much the population decreases (as a decimal),
- \( t \) is the number of years.

Let’s say we start with **50,000 llamas**. The population is going down by **10%** each year, which we can write as **0.10** in decimal form. We want to find out how many llamas there are after **10 years**.

So, we start by plugging our numbers into the formula:

\[
P(10) = 50000 \cdot (1 - 0.10)^{10}
\]

This simplifies to:

\[
P(10) = 50000 \cdot (0.90)^{10}
\]

Next, we need to calculate \( (0.90)^{10} \). When we do that, we find:

\[
(0.90)^{10} \approx 0.3487
\]

Now we can find out the new llama population:

\[
P(10) \approx 50000 \cdot 0.3487
\]

Calculating that gives us:

\[
P(10) \approx 17435
\]

So, after **10 years**, there would be about **17,435 llamas** left.