Problem 120 - College Algebra

Question

A town's population has been growing linearly. In \(2003,\) the population was \(45,000,\) and the population has been growing by 1,700 people each year. Write an equation, \(P(t),\) for the population \(t\) years after 2003 .

Step-by-Step Solution

Answer

The equation is \( P(t) = 45000 + 1700t \).

Step 1: Understand the Problem

The problem states that the population grows linearly. We need to express the population as a function of time. The function should include the initial population for the year 2003 and a growth rate of 1,700 people per year.

Step 2: Identify Initial Conditions

In 2003, the initial population, denoted as a starting point, is 45,000. This serves as the y-intercept or constant term in the linear equation.

Step 3: Determine the Slope

The population increases by 1,700 each year. This represents the slope or the rate of change in the equation. In context, each additional year ( ext{increases the population by this amount.)

Key Concepts

Population Modeling

Population modeling is a way of representing how a population changes over time. When we talk about linear growth in population modeling, it means the population increases at a constant rate each year. This is a simple but effective way to predict future population sizes if the conditions stay the same.

For instance:

• The given town's population started at 45,000 in 2003 and grows by 1,700 each year.
• This creates a straight line when you graph it over time, indicating steady growth.
• Population models can help us understand potential future issues, like resource needs.

It's essential to know that linear models assume no sudden changes in growth rates or factors affecting the population. This is why they are handy for short-term predictions but might not capture long-term effects accurately.

Overall, population modeling helps planners and policymakers prepare for the future by predicting how many people there will be.

Linear Equation

A linear equation is a mathematical statement used to describe a straight line. When applied to population modeling, it makes it easy to predict future values based on current trends. This particular type of equation shows how a population grows year by year.

The general form of a linear equation is \[ y = mx + b \]where:

• \( m \) = slope, which shows the rate of growth per year.
• \( b \) = y-intercept, representing the starting value in the initial year.

In our context:

• The slope \( m \) is 1,700, showing that each year, the population increases by 1,700.
• The y-intercept \( b \) is 45,000, indicating the initial population in 2003.

Putting it together, the linear equation for the town's population, \( P(t) = 1700t + 45000 \), shows how the population grows over time. Linear equations are great for capturing straightforward and consistent changes over time.

Rate of Change

The rate of change is a crucial concept in understanding linear growth models. It explains how quickly or slowly something is changing over a specific period. In population modeling, the rate of change tells us the increase or decrease in population in each time interval.

For instance, in the given town:

• The population grows by 1,700 people every year.
• This constant increase represents the rate of change, which remains the same every year.

A constant rate of change is characteristic of linear growth. It's a simple concept, yet understanding it can lead to valuable insights in predicting future changes.

Knowing the rate of change enables us to grasp the pace of growth or decline, making it easier to forecast future needs or allocate resources effectively. In our linear model, the constant rate of change simplifies projections, ensuring clarity and precision in predictions.