https://file.aiquickdraw.com/mathbot/file/1727408409557wei0hsfr.jpg

The rat population model is given by the equation:

\[
n(t) = 82e^{0.035t}
\]

Where:
- \( t \) represents the number of years since 1990.
- \( n(t) \) is the rat population in millions.

### 1. Rat Population in 1990
From the image, we can see the population for \( t = 0 \) (1990) is:

\[
n(0) = 82 \text{ million rats} = 82,000,000 \text{ rats}.
\]

This matches the given value of 82,000,000 rats in the image.

### 2. Rat Population in 2001
To find the rat population in 2001, we need to calculate \( n(t) \) for \( t = 2001 - 1990 = 11 \).

Using the formula:

\[
n(11) = 82e^{0.035 \times 11}
\]

Let me compute this for you.The model predicts that the rat population in the year 2001 was approximately **120,508,374 rats**.

Would you like more details or have any questions?

Here are five related questions to deepen your understanding:
1. How does the exponential model differ from a linear growth model in predicting populations?
2. What does the parameter 0.035 in the exponential model represent in practical terms?
3. How would the population change if the growth rate increased to 0.04?
4. Can this model predict when the rat population will double?
5. What assumptions does this model make about the environment affecting the rat population?

**Tip:** Exponential growth models can be sensitive to changes in the growth rate, leading to significant differences in long-term predictions.

What does the model predict the rat population was in the year 2001?

Discover how to predict the rat population using an exponential growth model with the formula n(t) = 82e^(0.035t). This example calculates the population for the year 2001. Suitable for students in Grades 9-12.

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