Problem 46 - Algebra 2

Question

For Exercises 46 and \(47,\) use the following information. From 1990 to \(2002,\) the number of births \(b(x)\) in the United States can be modeled by the function \(b(x)=-8 x+4045,\) and the number of deaths \(d(x)\) can be modeled by the function \(d(x)=24 x+2160,\) where \(x\) is the number of years since 1990 and \(b(x)\) and \(d(x)\) are in thousands. The net increase in population \(P\) is the number of births per year minus the number of deaths per year, or \(P=b-d\) . Write an expression that can be used to model the population increase in the U.S. from 1990 to 2002 in function notation.

Step-by-Step Solution

Answer

The population increase is modeled by \(P(x) = -32x + 1885\).

Step 1: Identify Given Functions

Begin by noting the functions given in the problem. The birth function is \(b(x) = -8x + 4045\), and the death function is \(d(x) = 24x + 2160\), where \(x\) represents the number of years since 1990.

Step 2: Understand Net Increase in Population

The net increase in population, \(P\), is defined as the number of births minus the number of deaths per year. Therefore, we need to subtract the function \(d(x)\) from the function \(b(x)\).

Step 3: Formulate the Population Increase Expression

To find the population increase expression, calculate \(P(x) = b(x) - d(x)\). Substitute the given functions: \(P(x) = (-8x + 4045) - (24x + 2160)\).

Step 4: Simplify the Expression

Simplify the expression for \(P(x)\). This involves distributing and combining like terms: \(P(x) = -8x + 4045 - 24x - 2160\). Therefore, \(P(x) = -32x + 1885\).

Key Concepts

Population Modeling

Population modeling is a mathematical representation used to describe dynamics related to population changes over time. When modeling populations, mathematicians and scientists use functions to predict future changes based on historical data. These models can be used to answer crucial questions about the growth, decline, or stability of a population over a specific period. By using linear functions, such as those provided in the exercise, we can model various aspects of population change.

• Linear functions are equations of the form \( y = mx + b \), where \(m\) is the slope and \(b\) is the y-intercept.
• In population modeling, the slope reflects the rate of change, indicating whether the population is increasing or decreasing, while the intercept represents the initial population size or starting point.

In the context of these exercises, the linear functions given describe both the birth and death rates in the U.S. from 1990 to 2002. This allows us to use a straightforward calculation to determine how the population has been affected by these factors.

Net Population Change

Net population change is determined by calculating the difference between births and deaths. This figure helps us understand whether a population is growing, shrinking, or remaining stable over time. The net change can be positive, which means more births than deaths, negative, which indicates more deaths than births, or zero, which shows equal births and deaths.

In mathematical terms, the net population change \(P(x)\) can be calculated using the difference between the births function \(b(x)\) and the deaths function \(d(x)\). Subtracting these functions gives us a clear view of the population trend:\[P(x) = b(x) - d(x)\]For the provided exercise:

• The births function: \(b(x) = -8x + 4045\)
• The deaths function: \(d(x) = 24x + 2160\)
• Therefore, \(P(x) = (-8x + 4045) - (24x + 2160)\) simplifies to \(P(x) = -32x + 1885\)

This equation provides a direct measurement of the population increase from 1990 to 2002.

Birth and Death Rates

Birth and death rates are crucial components of population studies, representing the number of births and deaths typically per 1,000 people per year. They directly influence how the overall population size changes over time.

• A higher birth rate compared to the death rate results in population growth.
• Conversely, a higher death rate than the birth rate will lead to a population decline.

In our exercise, birth rates and death rates are represented through linear functions:

• The birth rate model: \(b(x) = -8x + 4045\)
• The death rate model: \(d(x) = 24x + 2160\)

Each coefficient and constant in these equations is significant. For example:

• The coefficient \(-8\) in the birth function implies a slight decline in births over each year.
• The coefficient \(24\) in the death function indicates an increasing trend in deaths each year.

By analyzing these rates, researchers and policymakers can make informed decisions to address population-related challenges.