Problem 54 - College Algebra

Question

The June \(7,1999\) issue of Newsweek presented statistics showing progress African Americans have made in education, health, and finance. Infant mortality for African Americans is decreasing at a faster rate than it is for whites, shown by the graphs below. Infant mortality for African Americans can be modeled by \(M=-0.41 x+22\) and for whites by \(M=-0.18 x+10 .\) In both models, \(x\) is the number of years after 1980 and \(M\) is infant mortality, measured in deaths per 1000 live births. Use these models to project when infant mortality for African Americans and whites will be the same. What is infant mortality rate for both groups at that time? (GRAPH CAN'T COPY)

Step-by-Step Solution

Answer

The infant mortality rate for both African Americans and whites will be the same around the year 2033, and the common mortality rate approximately will be 0.46 deaths per 1000 live births at that time.

Step 1: Understand the Models

Each model represents the rate of infant mortality for different races. In each model, \(x\) represents the number of years after 1980, and \(M\) denotes the infant mortality rate (deaths per 1000 live births). The model for African Americans is \(M=-0.41x+22\) while for whites it's \(M=-0.18x+10\). The aim is to find the year when the mortality rates will be the same for both races.

Step 2: Set the Models Equal to Each Other

To find when these mortality rates will be the same, we need to solve for 'x' when the models are equal, meaning: \(-0.41x + 22 = -0.18x+10\).

Step 3: Solve for x

First, move the terms involving 'x' to one side and the constant terms to the other side to simplify the equation: \(-0.41x + 0.18x = 10 - 22\), which simplifies to \(-0.23x = -12\). Then, divide both sides by -0.23 to solve for 'x': \(x = -12/-0.23\) which is approximately \(x = 52.17\). Because 'x' represents years starting from 1980, we round 52.17 up to 53 (since we're dealing with time). So, they will be equal in the year \(1980+53 = 2033\).

Step 4: Calculate the Infant Mortality Rate

Once we have the year, we can find the infant mortality rate at that time by substituting 'x' in any of the two models because the rates will be equal. Let's substitute 'x' into the model for whites: \(M = -0.18*53 + 10\) which gives approximately \(M = 0.46\).

Key Concepts

Linear Equations

Linear equations are foundational concepts in algebra that express relationships with consistent rates of change. They can be used to model various real-world situations, such as the change in infant mortality rates. A linear equation typically has the form \(y = mx + b\), where:

• \(m\) is the slope, representing the rate of change of one variable with respect to another.
• \(b\) is the y-intercept, which is the value of \(y\) when \(x = 0\).
• \(x\) and \(y\) are the variables of interest.

In the case of infant mortality models, these linear equations use \(x\) to represent years after 1980. Meanwhile, \(M\), the infant mortality rate, is influenced by a negative slope, indicating a decrease in mortality over time. Understanding this helps us predict future trends and make informed decisions based on those predictions.

Intersection of Lines

The point where two lines intersect on a graph presents an important relationship between the corresponding linear models. To find the intersection of lines, we set the equations equal to each other. This approach is particularly useful in scenarios where we need to determine when two different outcomes will equate, such as the equalization of infant mortality rates for African Americans and whites.
For the given problem, we equate the models for African Americans and whites:\[-0.41x + 22 = -0.18x + 10\]This calculation finds the value of \(x\) where both models project identical mortality rates, marking an important intersection that allows for insightful comparisons of different demographic trends over time.

Algebraic Solutions

Resolving real-world questions with algebra involves using operations to isolate variables and find solutions. In the case of infant mortality models, solving the intersection involves simplifying the equation to find \(x\).

Starting with this equation:\[-0.41x + 22 = -0.18x + 10\]We move terms to isolate \(x\):\[-0.41x + 0.18x = 10 - 22 \-0.23x = -12\]Finally, solving for \(x\) involves dividing both sides by \(-0.23\):\[x = \frac{-12}{-0.23} \approx 52.17\]Rounding up to the next full year, since time cannot be fractional in this context, tells us the year of equal mortality rates is 2033. Adeptly maneuvering through these calculations requires understanding and applying algebraic principles.

Graphical Analysis

Graphical analysis offers a visual representation of mathematical models, allowing for an intuitive understanding of data trends and intersections. By plotting the linear equations for infant mortality rates on a graph, students can easily observe where these lines intersect, which would indicate the year both mortality rates are the same.

Graphical analysis can highlight:

• Trends over time, showing how each group's rate decreases.
• The specific point (year) of intersection, offering a powerful visual tool for comparing rates.
• The impacts of varying slopes on the rate of decline in different demographics.

Understanding graphs can make it easier for students to comprehend complex algebraic solutions, cementing the relationship between numerical data and its real-world implications.