Problem 88 - College Algebra with Modeling and Visualization

Question

Population Density The population density \(D\) of the United States in people per square mile during year \(x\) from 1900 to 2000 can be approximated by the formula \(D(x)=0.58 x-1080 . \quad\) (Source: Bureau of the Census.) (a) Interpret the slope of the graph of \(D\). (b) Estimate when the density varied between 50 and 75 people per square mile.

Step-by-Step Solution

Answer

The slope is 0.58 people per year. Density ranged from 1949 to 1991.

Step 1: Interpret the Slope

The given formula for population density is \( D(x) = 0.58x - 1080 \). Here, the slope is \( 0.58 \). The slope indicates the rate at which the population density is increasing yearly. Therefore, we interpret the slope as the average annual increase in population density, which is 0.58 people per square mile per year.

Step 2: Set Up Inequality for the Density Range

To find when the population density varies between 50 and 75 people per square mile, we set up the inequality: \[ 50 \leq 0.58x - 1080 \leq 75 \]

Step 3: Solve the Lower Bound Inequality

Start with the lower bound of the inequality: \[ 50 \leq 0.58x - 1080 \] Add 1080 to both sides:\[ 1130 \leq 0.58x \] Now, divide by 0.58 to solve for \( x \):\[ x \geq \frac{1130}{0.58} \approx 1948.28 \].Rounding up, we find \( x \geq 1949 \).

Step 4: Solve the Upper Bound Inequality

Now, solve the upper bound of the inequality:\[ 0.58x - 1080 \leq 75 \] Add 1080 to both sides:\[ 0.58x \leq 1155 \] Divide by 0.58:\[ x \leq \frac{1155}{0.58} \approx 1991.38 \].Rounding down, we find \( x \leq 1991 \).

Step 5: Combine Results

Combining the results of the previous steps, the population density varied between 50 and 75 people per square mile from approximately 1949 to 1991.

Key Concepts

Slope Interpretation

In mathematics, the slope of a line tells us how steep the line is. It's an important concept in various applications, including understanding population density. When looking at the formula for population density, \( D(x) = 0.58x - 1080 \), the slope is represented by the number 0.58. This indicates how much the population density changes per year.

In the context of this formula, the slope of 0.58 signifies that every year, the population density increases by 0.58 people per square mile. So, if you imagine a graph, with years on the x-axis and population density on the y-axis, the line would rise slowly depicting this increase. Knowing the slope helps predict future population trends and understand past growth.

Because the slope is a positive number, it indicates a steady growth in density. It's also part of a linear function, meaning the rate of increase is constant over time.

Inequality Solving

Solving inequalities is a fundamental algebraic skill, useful in finding ranges for values within constraints. In this exercise, we're dealing with the population density ranging between 50 and 75 people per square mile.

The inequality setup looks like this:

• \( 50 \leq 0.58x - 1080 \)
• \( 0.58x - 1080 \leq 75 \)

For both parts, you need to solve for \( x \), the year.

The process involves:

• Adding 1080 to each part of the inequality to isolate terms involving \( x \).
• Then, dividing each result by 0.58 to solve for \( x \).

The solutions show the specific years when the population density stays within the range we are interested in. Solving inequalities is a crucial procedure to define precise intervals or limits in a dataset.

Mathematical Modeling

Mathematical modeling involves using mathematical concepts and structures to describe real-world scenarios. The formula for population density, \( D(x) = 0.58x - 1080 \), is an example of such a model.

This model simplifies complex data into a linear form, making it easier to predict and understand trends. Here, the variable \( x \) stands for the year, and \( D(x) \) for population density. The linear model assumes a steady rate of change, denoted by the slope.

Mathematical models are essential because:

• They provide a framework to study how one quantity affects another. In this case, how years influence density.
• These models allow for easy scalability to make predictions about future data points.

It's key to remember that models are approximations; reality may introduce variations, but they provide a useful tool for initial understanding and analysis.

Algebraic Functions

An algebraic function is a relationship between variables that use algebraic operations like addition, subtraction, multiplication, and division. The given population density function, \( D(x) = 0.58x - 1080 \), is a classic example of a linear algebraic function.

Understanding such functions involves:

• Identifying variables: \( x \) is the independent variable (year), and \( D(x) \) the dependent one (density).
• Recognizing the constant rate of change (the slope).
• Solving the function to find values of \( x \) that meet certain conditions, as seen in solving inequalities.

These functions provide clarity in visualizing how changes in one variable directly affect another.

The linear nature implies straightforward calculations, making algebraic functions a crucial tool in analyzing and applying mathematics to real-life problems, such as understanding changes in population density over time.