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 96.

x

0

9

25

30

46

51

70

76

P(x)

9

37

106

135

196

218

290

310

Remember: when predicting a population, always discard partial units dash– in other words, round down.

To determine a linear model and predict the population in year 96, we'll follow these steps:

### Given Data:

\[
\begin{array}{c|c|c|c|c|c|c|c|c}
x & 0 & 9 & 25 & 30 & 46 & 51 & 70 & 76 \\
\hline
P(x) & 9 & 37 & 106 & 135 & 196 & 218 & 290 & 310 \\
\end{array}
\]

### Step 1: Select Two Points
To create a linear model, we'll select two points from the data. We typically choose the earliest and latest data points to best capture the overall trend. Here, we'll use the points \((0, 9)\) and \((76, 310)\).

### Step 2: Calculate the Slope (\(m\))
The slope \(m\) is calculated using the formula:

\[
m = \frac{P(x_2) - P(x_1)}{x_2 - x_1} = \frac{310 - 9}{76 - 0} = \frac{301}{76} \approx 3.9618
\]

### Step 3: Write the Equation of the Line
The linear model is in the form \(P(x) = mx + b\). We already have \(m \approx 3.9618\). Now, we'll solve for \(b\) using the point \((0, 9)\):

\[
9 = 3.9618(0) + b \quad \Rightarrow \quad b = 9
\]

So the equation becomes:

\[
P(x) = 3.9618x + 9
\]

### Step 4: Predict the Population for Year 96
Substitute \(x = 96\) into the linear model:

\[
P(96) = 3.9618(96) + 9 = 380.3328 + 9 = 389.3328
\]

### Step 5: Round Down to Discard Partial Units
Since we round down:

\[
\text{Predicted population in year 96} = 389
\]

### Final Answer:
The predicted population in year 96 is **389**.

Do you have any questions, or would you like further details?

Here are some related questions:

1. How would the prediction change if we used different points for the linear model?
2. What is the significance of the slope in this context?
3. How do you interpret the intercept in the linear model?
4. What are the limitations of using a linear model for this prediction?
5. How would you check if a linear model is the best fit for the data?

**Tip:** Always verify if the linear model fits well by checking the residuals (differences between actual and predicted values) across the data points!

Learn how to predict the population of a species in year 96 using a linear model. Follow a detailed step-by-step solution with data analysis and mathematical formulas. Suitable for students and professionals interested in data analysis and mathematical modeling.

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