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!