Question

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.

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Answer 1

To figure out how many llamas are left after a certain number of years when the population is slowly going down, 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 how many llamas we started with,
- \( r \) is the rate at which 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 know 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.