Problem 38 - College Algebra

Question

For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs shows the population and the year over the ten-year span, (population, year) for specific recorded years: \((2500,2000),(2650,2001),(3000,2003),(3500,2006),(4200,2010)\) Predict when the population will hit \(8,000\) .

Step-by-Step Solution

Answer

The population will hit 8,000 in the year 2033.

Step 1: Define the Linear Equation

We need to find the equation of the line representing the population growth over time. We'll assume a linear model given the steady increase in population. A linear equation has the general form \( y = mx + b \), where \( y \) is the population, \( x \) is the year, \( m \) is the slope, and \( b \) is the y-intercept.

Step 2: Calculate the Slope (m)

To find the slope \( m \), we use the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Using two points from the given data, for example, \((2500, 2000)\) and \((4200, 2010)\), we calculate:\[m = \frac{4200 - 2500}{2010 - 2000} = \frac{1700}{10} = 170\]This means the population increases by 170 each year.

Step 3: Calculate the Y-Intercept (b)

Knowing that \( y = mx + b \) and using one of the points, say \((2500, 2000)\), we substitute \( m = 170 \), \( y = 2500 \), \( x = 2000 \) to find \( b \):\[2500 = 170 \times 2000 + b\]Solving for \( b \):\[2500 = 340000 + b \Rightarrow b = 2500 - 340000 = -337500\]

Step 4: Form the Linear Equation

Now we can form the population model using the slope \( m = 170 \) and y-intercept \( b = -337500 \):\[y = 170x - 337500\]This is the equation that describes the population over time.

Step 5: Solve for Year When Population is 8000

To find when the population will reach 8000, substitute \( y = 8000 \) into the equation and solve for \( x \):\[8000 = 170x - 337500\]\[8000 + 337500 = 170x\]\[345500 = 170x\]\[\ x = \frac{345500}{170} = 2032.35\]So, the population will reach 8000 in the year 2033.

Key Concepts

Population Growth

Population growth refers to the change in the number of inhabitants of a city or any specific area over time. In our scenario, we're observing a steady increase in population over a ten-year period.

Linear equations are ideal for modeling situations like this because they assume that changes happen at a constant rate. This is why we use a linear equation to track population changes over time.

Understanding population growth is crucial in fields like urban planning and resource management. It helps in predicting future needs for infrastructure, schools, and hospitals. Ultimately, this information aids in strategic decision-making to improve living conditions.

Slope Calculation

The slope is a measure of how steep a line is, or in this context, how rapidly the population is growing. In the formula of a linear equation, expressed as \( y = mx + b \), the slope is represented by \( m \).

To calculate the slope, we use the change in the population over the change in years between two data points. The formula for slope calculation is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For example, using the points \((2500, 2000)\) and \((4200, 2010)\), the slope is:\[ m = \frac{4200 - 2500}{2010 - 2000} = \frac{1700}{10} = 170 \]This indicates that the population grows by 170 individuals each year.

A correct understanding of slope helps us to appreciate how changes in one variable can affect another, highlighting trends in data.

Y-Intercept

The y-intercept in a linear equation is the point where the line crosses the y-axis when the value of \( x \) is zero. It is represented by \( b \) in the equation \( y = mx + b \).

To find the y-intercept, you can use the calculated slope and one of the data points provided in the problem. Using the point \((2500, 2000)\) and the slope \( m = 170 \), we solve for \( b \) as follows:

• Substitute \( x = 2000 \), \( y = 2500 \), and \( m = 170 \) into the equation \( y = mx + b \).
• This gives: \( 2500 = 170 \times 2000 + b \).
• Solve for \( b \) to get: \( b = -337500 \).

The y-intercept helps us understand where a trend line will begin if we backtrack to when \( x \) equals zero, although it might not always make practical sense to do so.

Prediction of Future Values

Predicting future values involves using the linear equation model to estimate what will happen at a specific time in the future. For population growth, this means determining when the population will reach a certain number.

In our example, to find out when the population hits 8000, we substitute \( y = 8000 \) into the linear equation \( y = 170x - 337500 \) and solve for \( x \).

Here's how you do it:

• Start with the equation: \( 8000 = 170x - 337500 \).
• Add 337500 to both sides to get: \( 8000 + 337500 = 170x \).
• Culminate in: \( 345500 = 170x \).
• Divide both sides by 170 to solve for \( x \), which results in \( x = 2032.35 \).

Rounding up, we predict the population will reach 8000 in the year 2033. This method of prediction is valuable in planning and preparing for future changes or needs.