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Chapter 6: Problem 86

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. $$\begin{aligned}&y=0.0515 x+12.3\\\&y=0.255 x+9.01\end{aligned}$$ In each case, \(x\) represents the number of years since 1990 . (Source: U.S. Census Bureau.) (a) Solve the system to find the year when these population percents were equal. (b) What percent of the U.S. resident population will be 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?

Short Answer

Expert verified
(a) Year 2006; (b) 13.09%; (d) Spanish/Hispanic/Latino population.

Step by step solution

01

Set Equations Equal

To find when the population percentages are equal, set the two equations equal to each other: \[ 0.0515x + 12.3 = 0.255x + 9.01 \]
02

Simplify the Equation

Simplify the equation by subtracting \(0.0515x\) and \(9.01\) from both sides: \[ 12.3 - 9.01 = 0.255x - 0.0515x \]\[ 3.29 = 0.2035x \]
03

Solve for x

Divide both sides by 0.2035 to find \(x\): \[ x = \frac{3.29}{0.2035} \approx 16.17 \] Round \(x\) to 16 as we calculate for actual years.
04

Calculate the Year

Add \(x = 16\) to 1990 to find the year: \[ 1990 + 16 = 2006 \] Thus, the two populations were equal in 2006.
05

Calculate Hispanic/Latino Percentage in 2006

Substitute \(x = 16\) into either equation, using \[ y = 0.255x + 9.01 \] Calculate as follows: \[ y = 0.255(16) + 9.01 = 4.08 + 9.01 = 13.09 \] Therefore, in 2006, the Spanish/Hispanic/Latino population was 13.09%.
06

Graph the Equations

Plot both equations on the same graph. Use separate colors or styles to differentiate Black and Spanish/Hispanic/Latino populations. The point of intersection at \(x = 16\) (or the year 2006) confirms the analytical solution.
07

Determine the Faster Increasing Population

Compare the slopes of the equations, \(0.0515\) for Black and \(0.255\) for Spanish/Hispanic/Latino. A greater slope indicates a faster increase; thus, the Spanish/Hispanic/Latino population is increasing more rapidly.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population Modeling
Population modeling allows us to construct predictions about how demographic groups will grow over time. It uses mathematical equations to anticipate changes. This approach is incredibly useful for planning resources, managing community services, and shaping policies.
Models can be as simple as linear equations, where growth is steady over time. This means the population percentage increases or decreases at a constant rate. In the given exercise, the equations forecast the population percentages of Black and Spanish/Hispanic/Latino people in the U.S. from 1990 to 2050. Here, time (\(x\)) varies, representing years since 1990, and allows us to calculate percentages (\(y\)).
Understanding these models helps convey valuable insights regarding demographic trends. These insights assist governments, organizations, and researchers in making informed decisions.
  • Linear population models assume constant growth rates.
  • They are easy to calculate and provide useful approximations for short-term predictions.
While linear models are basic, they form the groundwork for more complex analyses, such as incorporating factors like migration, birth rates, and economic influences.
System of Equations
A system of equations involves multiple equations containing multiple variables. Our exercise uses a system of two linear equations, each representing a population growth model. The variables, in this case, are the years since 1990 and the projected population percent.
By solving this system, we identify the point where the two populations share the same percentage. This is done by setting the equations equal and solving for the variable, \(x\), which shows the years since 1990 when the levels equalize.
This exercise demonstrates solving systems analytically and graphically:
  • Analytically: Set the equations equal to find the intersection point.
  • Graphically: Plot both lines and see where they meet visually, confirming the analytical result.
Systems of equations are crucial in many fields because they offer solutions to complex problems described by numerous variables. They allow us to explore relationships, correlations, and forecasts through different perspectives.
Rate of Change
Rate of change is a fundamental concept in calculus and algebra, representing how a quantity changes concerning another. In linear equations, it equates to the slope of the line. The slope signifies how quickly or slowly one variable changes as the other progresses.
In this exercise, the slope (\(m\)) in each equation (\(0.0515\) and \(0.255\)) represents how much the population percentage increases per year since 1990. A steeper slope indicates a faster rate of change and thus a quicker growth rate.
To determine which population is growing faster, we directly compare slopes:
  • The Hispanic/Latino population has a higher slope (\(0.255\)), indicating it grows faster compared to the Black population (\(0.0515\)).
The concept of rate of change is applied widely, not only in population studies but also in finance, physics, and any field that analyses temporal data. It offers insights into trends and helps compare different scenarios.

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Most popular questions from this chapter

The break-even point for a company is the point where costs equal revenues. If both cost and revenue are expressed as linear equations, the break-even point is the solution of a linear system. In each exercise, \(C\) represents cost in dollars to produce x items, and R represents revenue in dollars from the sale of \(x\) items. Use the substitution method to find the break-even point in each case-that is, the point where \(C=R .\) Then find the value of \(C\) and \(R\) at that point. $$\begin{aligned}&C=20 x+10,000\\\&R=30 x-11,000\end{aligned}$$

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