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The problem in the image asks to determine a linear model and predict the population in year 94, based on a given table of population data at specific time points.

### Given Table:
- \( P(x) \) represents the population at time \( x \) (in years).
- The table includes these values:

\[
\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline
x & 0 & 10 & 17 & 24 & 42 & 59 & 71 \\
\hline
P(x) & 0 & 115 & 243 & 292 & 385 & 401 & 501 \\
\hline
\end{array}
\]

### Task:
- Find a linear model to fit the data.
- Predict the population at \( x = 94 \) using this model.
- Round down to the nearest whole number when predicting the population.

Let me work through this problem step by step by calculating the linear regression model from the data. I'll then use the model to predict the population for \( x = 94 \).The predicted population for year 94, based on the linear model, is approximately **666** (rounded down as instructed).

Would you like further details or have any other questions? Here are some follow-up questions:

1. How is a linear model determined using data points?
2. What does the slope of the linear model represent in this context?
3. Can linear models always accurately predict populations for large time spans?
4. How would this prediction change with a different modeling technique, such as quadratic regression?
5. Why is it important to round down in population predictions?

**Tip:** When using linear models, always check the residuals (differences between predicted and actual values) to assess the model's accuracy.

Determine a linear model and predict the population in year 94, based on the given table of population data at specific time points.

This problem involves using a linear model to predict the population in year 94 based on the provided population data at specific time points. The solution includes using linear regression to fit a line and make the prediction, rounding down the final answer as required.

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