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:

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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GPT-4o mini · Answer

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.