Consider this scenario: A town's population has been decreasing at a constant rate. In 2010 the population was 5,100. By 2013 the population had dropped to 3,300. Assume this trend continues.

Predict the population in 2016.

To solve this problem, we can model the population decrease using a linear equation because the population is decreasing at a constant rate. Here's the step-by-step process:

### Step 1: Define Variables
Let \( P(t) \) represent the population at year \( t \). We know the population values for two points in time:
- In 2010 (\( t = 2010 \)), the population was 5,100.
- In 2013 (\( t = 2013 \)), the population was 3,300.

### Step 2: Determine the Rate of Decrease
The rate of decrease (slope, \( m \)) can be found using the formula for the slope between two points \((t_1, P_1)\) and \((t_2, P_2)\):

\[
m = \frac{P_2 - P_1}{t_2 - t_1} = \frac{3300 - 5100}{2013 - 2010} = \frac{-1800}{3} = -600
\]

This tells us the population decreases by 600 people per year.

### Step 3: Write the Linear Equation
The linear equation for the population can be written as:

\[
P(t) = m(t - t_1) + P_1
\]

Substituting \( m = -600 \), \( t_1 = 2010 \), and \( P_1 = 5100 \):

\[
P(t) = -600(t - 2010) + 5100
\]

### Step 4: Predict the Population in 2016
Now, substitute \( t = 2016 \):

\[
P(2016) = -600(2016 - 2010) + 5100 = -600 \times 6 + 5100 = -3600 + 5100 = 1500
\]

So, the predicted population in 2016 is **1,500**.

Would you like further details or have any questions?

### Related Questions:
1. What would be the population in 2020 if the trend continues?
2. How long will it take for the population to reach zero at this rate?
3. Can you represent the population decrease on a graph?
4. What is the population rate of decrease as a percentage per year?
5. How would the prediction change if the decrease rate was not constant?

**Tip:** When working with linear models, always identify the slope first as it tells you how fast the quantity is changing over time!

Consider this scenario: A town's population has been decreasing at a constant rate. In 2010 the population was 5,100. By 2013 the population had dropped to 3,300. Assume this trend continues. Predict the population in 2016.

Learn how to predict a town's population in 2016 using linear equations based on a constant rate of decrease. This problem involves key mathematical concepts such as slope-intercept form and population modeling, suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation