Problem 100 - A Graphical Approach to Precalculus with Limits

Question

Populations of Minorities in the United States \(\quad\) The current and estimated resident populations, \(y\) (in percent), of Black and Spanish/Hispanic/Latino people in the United States for the years \(1990-2050\) are modeled by the following linear equations. $$ \begin{array}{ll} y=0.0515 x+12.3 & \text { Black } \\ y=0.255 x+9.01 & \text { SpM } \end{array} $$ Sp.Hisp \(\Omega\) at. In each case, \(x\) represents the number of years after 1990 . (a) Solve the system to find the year when these population percents were equal. (b) What percent of the U.S. resident population was Spanish/Hispanic/Latino in the year found in part (a)? (c) Graphically support the analytic solution in part (a). (d) Which population is increasing more rapidly?

Step-by-Step Solution

Answer

The populations were equal around 2006; Spanish/Hispanic/Latino was ~13.67%. Hispanic/Latino is increasing more rapidly.

Step 1: Set Up the Equation for Equality

To find the year when the population percentages were equal, set the two equations equal to each other. This means setting \(0.0515x + 12.3 = 0.255x + 9.01\).

Step 2: Solve for x

Subtract \(0.0515x\) from both sides to get \(12.3 = 0.2035x + 9.01\). Then, subtract 9.01 from both sides to get \(3.29 = 0.2035x\). Finally, divide both sides by 0.2035 to find \(x = \frac{3.29}{0.2035}\), which is approximately \(16.17\).

Step 3: Calculate the Year

Since \(x\) represents the number of years after 1990, add 16.17 to 1990 to find the year. The year is approximately 2006.

Step 4: Determine the Spanish/Hispanic/Latino Percent in 2006

Substitute \(x = 16.17\) into the Spanish/Hispanic/Latino equation, \(y = 0.255x + 9.01\), to find \(y \approx 0.255 \times 16.17 + 9.01\). Calculate this to find \(y \approx 13.67\%\).

Step 5: Identify More Rapid Increase

Compare the slopes of the two equations: 0.0515 for Black and 0.255 for Spanish/Hispanic/Latino. The slope for Spanish/Hispanic/Latino is higher, indicating a more rapid increase.

Key Concepts

Population Projections

Population projections are a vital tool used to predict future population trends based on historical data. In this exercise, the population percentages of Black and Spanish/Hispanic/Latino communities in the United States are modeled using linear equations. The goal is to estimate these percentages for various future years, specifically from 1990 to 2050.
Linear equations express the relationship between time and population percentage. The variable \(x\) represents the number of years after 1990, allowing us to track changes over time:

• For Black populations, the equation is \(y=0.0515x+12.3\).
• For Spanish/Hispanic/Latino populations, the equation is \(y=0.255x+9.01\).

Using these equations, we can calculate estimated potential shifts in demographics, predict when these populations might reach certain thresholds, and identify key changes. Understanding how to use these projections helps in planning for economic, social, and infrastructural needs.
Population projections are indispensable for government planning, policy-making, and resource allocation to ensure communities' needs are met as demographics evolve.

Graphical Analysis

Graphical analysis acts as a visual tool that helps us understand and verify the results of linear equations for population predictions. In the exercise, we've solved linear equations analytically. However, graphical analysis provides a visual check, enhancing clarity.
By graphing each equation on a coordinate plane, you can observe where the lines intersect. This intersection symbolizes the year when the population percentages of Black and Spanish/Hispanic/Latino populations are equal.
To graph these equations, plot \(y=0.0515x+12.3\) and \(y=0.255x+9.01\), with \(x\) as the horizontal axis (years since 1990) and \(y\) as the vertical axis (percentage of population). The intersection point revealed on the graph corresponds to the year 2006, aligning with analytic calculations.
This visual representation makes it easier to comprehend both the long-term trends and changes over time. Graphical analysis is not only supportive but crucial for confirming mathematical solutions and understanding potential impacts on society.

Rate of Change in Populations

The concept of rate of change in populations provides insights into how quickly a population percentage is increasing or decreasing over time. It's crucial for understanding demographic trends and planning accordingly.
In this exercise, the rate of change is represented by the slope of the linear equation, which indicates how the population percentage varies with every passing year. For population growth, higher slope values signify faster growth.

• The slope of the Black population equation is 0.0515, showing a gradual increase.
• The slope for the Spanish/Hispanic/Latino population is 0.255, indicating a more rapid rise.

This difference highlights that the Spanish/Hispanic/Latino community is growing at a significantly faster rate compared to the Black community. By understanding these rates, policy-makers and demographers can allocate resources where they're needed most and better prepare for future societal changes.
The rate of change is a powerful metric that helps anticipate which demographic groups may need more attention and support based on growth trends.