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 88.
x	0	11	25	31	44	59	66	74	P(x)	2	58	122	149	227	302	331	367
Remember: when predicting a population, always discard partial units dash – in other words, round down.

To determine a linear model for the population \( P(x) \) over time \( x \), we'll first calculate the best-fitting line using the given data points. The linear model has the form:

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

Where:
- \( m \) is the slope of the line.
- \( b \) is the y-intercept.

### Step 1: Calculate the Slope \( m \)
The slope \( m \) is calculated using the formula:

\[
m = \frac{n\sum{(xy)} - \sum{x}\sum{y}}{n\sum{x^2} - (\sum{x})^2}
\]

### Step 2: Calculate the Intercept \( b \)
The y-intercept \( b \) is calculated using:

\[
b = \frac{\sum{y} - m\sum{x}}{n}
\]

Where \( n \) is the number of data points, \( x \) values are the time in years, and \( y = P(x) \) are the corresponding populations.

### Step 3: Use the Linear Model to Predict the Population at Year 88
After calculating the slope and intercept, substitute \( x = 88 \) into the linear equation to predict the population.

I'll proceed with the calculations to determine the slope \( m \) and intercept \( b \).The linear model predicts that the population at year 88 would be approximately 441. Since you mentioned to discard partial units, the predicted population in year 88 is **441**.

Would you like further details or have any questions?

Here are 5 related questions:
1. How is the slope \( m \) of a linear model interpreted in this context?
2. What could be the limitations of using a linear model for predicting population growth?
3. How would you check the accuracy of the linear model?
4. Can you explain how to determine if a linear model is the best fit for this data?
5. What would be the population in year 100 using this model?

**Tip:** When fitting a model to data, always check the residuals (differences between observed and predicted values) to ensure that the model appropriately captures the data trend.

Learn how to predict population using a linear model based on given data points. Calculate slope and intercept, apply the model to predict population in year 88. Detailed solution and formula application provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation