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To determine a linear model from the given data, we need to approximate the relationship between \(x\) (time in years) and \(P(x)\) (population). We'll use linear regression to find the best-fitting line, which takes the form:

\[
P(x) = mx + b
\]

Where:
- \(m\) is the slope of the line (rate of change of population),
- \(b\) is the y-intercept (population at \(x = 0\)).

We can then use this model to predict the population at \(x = 86\). Let me calculate this for you.The linear model for the population can be expressed as:

\[
P(x) = 7.05x + 2.94
\]

Using this model, the predicted population at year \(x = 86\) is approximately:

\[
P(86) = 7.05(86) + 2.94 \approx 609.54
\]

Since we are instructed to round down, the predicted population in year 86 is **609**.

Would you like a more detailed explanation or have any questions? Here are some related questions:

1. How is the slope in a linear regression interpreted?
2. Can the linear model predict population beyond the given data range?
3. How would you calculate the population if the relationship was non-linear?
4. What impact does rounding down have on long-term predictions?
5. Could an exponential model fit this data better?

**Tip**: When working with population data, always check if a linear model is appropriate or if a different model (e.g., exponential) might provide better accuracy!

The data in the table shows the population P(x) of a particular species at time x (in years). Determine a linear model and predict the population in year 86.

Learn how to determine a linear model using population data and predict future populations. This example walks through finding the best-fitting line and using it to estimate the population at year 86. Suitable for students in Grades 9-12.

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