Problem 10 - College Algebra

Question

For the following exercises, consider this scenario: A town's population has been decreasing at a constant rate. In 2010 the population was \(5,900.5 \mathrm{y} 2012\) the population had dropped \(4,700 .\) Assume this trend continues. Identify the year in which the population will reach 0.

Step-by-Step Solution

Answer

The town's population will reach 0 towards the end of 2019.

Step 1: Identify the Known Variables

We know the population in 2010 was 5900 and in 2012 it was 4700. These give us two data points: \((2010, 5900)\) and \((2012, 4700)\).

Step 2: Calculate the Rate of Change

To find the rate of change, calculate the decrease in population over the change in years. The population decreased by \(5900 - 4700 = 1200\) over \(2012 - 2010 = 2\) years. So, the rate of change is \(\frac{1200}{2} = 600\) people per year.

Step 3: Establish a Linear Equation

We can consider this as a linear equation \( y = mx + b \), where \( m \) is the rate of change and \( b \) is the initial population from 2010. Thus, \( y = -600x + 5900 \).

Step 4: Set the Equation to Zero to Find the Year

To find when the population reaches 0, set \( y = 0 \) in the equation: \( 0 = -600x + 5900 \). Solve for \( x \) to find the number of years since 2010 when the population will be 0.

Step 5: Solve for x

Add 600x to both sides to obtain \( 600x = 5900 \). Then divide by 600: \( x = \frac{5900}{600} \approx 9.83 \).

Step 6: Determine the Year

Since \( x \) is approximately 9.83, and \( x = 0 \) represents the year 2010, add 9.83 to 2010 to find the year in which the population reaches 0: \( 2010 + 9.83 \approx 2019.83 \). This means the population will reach 0 towards the end of 2019.

Key Concepts

Rate of Change

The concept of "Rate of Change" is very handy when dealing with situations where a quantity changes over time. In the context of a linear equation, the rate of change is essentially the 'slope' of the line.
For the problem at hand, we talked about the population of a town decreasing at a constant rate. The rate of change in this scenario tells us how many people are leaving per year.

• In our exercise, the town's population dropped from 5,900 in 2010 to 4,700 in 2012.
• To find the rate of change, we calculate the difference in population and divide it by the number of years.
• This gives us a rate of 600 people per year.

The negative sign is important because it indicates a decrease in population. This tells us that every year, 600 fewer people are in the town. Grasping this change helps us make predictions about future trends.

Population Model

Population models like the one in this exercise help us understand how a population grows or decreases over time. This model assumes a constant rate of change, which simplifies the real-world complexities.
In our town's scenario, the model effectively predicts future population levels by using a linear equation:

• The initial population, given as 5,900 in 2010, serves as our starting 'point,' often known as the 'y-intercept' in algebraic terms.
• The linear equation derived: \( y = -600x + 5900 \), models the population, with \( y \) as population and \( x \) as the number of years since 2010.

This equation allows us to insert any value for \( x \) to estimate the population for particular years. With this model, we predict when the population may reach zero, identifying significant turning points.

Algebraic Problem Solving

Algebraic problem solving plays a crucial role in unpacking scenarios like the one in this exercise. By utilizing linear equations and valuable algebraic techniques, we address many real-world problems effectively.
Let's break it down:

• First, you're given specific data points for years and population.
• These points are used to calculate the rate of change, as described as our step 2 in the solution.
• Next, a linear equation is established using this rate and initial population figures: \( y = -600x + 5900 \).

The final step involves solving the equation for a specific condition—in this case, when the population reaches zero.

• You set \( y = 0 \) and solve for \( x \).
• The solution shows that approximately 9.83 years after 2010, the population will reach zero.
• Add this to the base year 2010 to estimate the timeline, predicting around late 2019.

Through simple calculations and logical steps, algebra makes what seems complicated quite manageable. It's a powerful tool for prediction and understanding various trends.