Problem 31 - College Algebra and Calculus: An Applied Approach

Question

The population \(P\) (in millions) of the United States from 1992 through 2005 can be modeled by the exponential function \(P(t)=252.12(1.011)^{\prime}\), where \(t\) is the time in years, with \(t=2\) corresponding to 1992 . Use the model to estimate the population in the years (a) 2008 and (b) 2012. (Source: U.S. Census Bureau)

Step-by-Step Solution

Answer

By computing the given exponential function using the relevant \(t\)-values, it's estimated that the population of the United States was about 300.75 million in 2008 and about 331.62 million in 2012.

Step 1: Find the value of \(t\) for 2008

Since \(t = 2\) represents the year 1992, we need to compute how many years there are from 1992 to 2008. That is \(2008 - 1992 = 16\). So, \(t = 16\) for the year 2008.

Step 2: Substitute \(t = 16\) into the model

We substitute \(t = 16\) into \(P(t)=252.12(1.011)^t\). This gives \(P(16)=252.12(1.011)^{16}\). We can then compute this equation to get an estimate.

Step 3: Find the value of \(t\) for 2012

It's a similar process for the year 2012. We need to compute how many years there are from 1992 to 2012. That is \(2012 - 1992 = 20\). So, \(t = 20\) for the year 2012.

Step 4: Substitute \(t = 20\) into the model

Substitute \(t = 20\) into \(P(t)=252.12(1.011)^t\). This gives \(P(20)=252.12(1.011)^{20}\). We can compute this equation to get the estimate for 2012.

Key Concepts

Population Modeling

Population modeling is a mathematical approach used to predict how a population will grow over time. Imagine we're playing with numbers that represent people to see how their total might change in the future. One of the most commonly used models is the exponential growth model. This is often used when population changes quickly, as seen in many real-world scenarios.

In the context of population studies, these models take several factors into account:

• Birth and death rates
• Migration patterns
• Economic influences

For example, using the model provided, we estimate the population of the United States in future years by applying mathematics that describes how populations tend to grow. In our specific exercise, the population in any given year since 1992 can be estimated using an equation steeped in exponential principles. This allows planners and governments to prepare for future changes.

Exponential Functions

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. This might sound complex but is quite powerful. We use exponential functions to model processes where change happens at a constant percentage rate.

In our population modeling problem, the function used is \(P(t) = 252.12(1.011)^t\). Let’s break it down:

• \(252.12\) represents the initial population size in millions when \(t = 2\).
• \(1.011\) is the growth factor, indicating the rate at which the population is expanding. A value greater than 1 suggests growth.

Exponential functions capture the essence of real-world growth scenarios, where the rate of increase is proportional to the current amount. This makes them applicable to fields beyond population modeling, like finance and physics.

Algebra

Algebra serves as the backbone for solving equations related to population modeling and exponential functions. It gives us the tools to manipulate and solve equations like the ones we encounter in this population exercise. Through understanding algebra, we can substitute values, simplify expressions, and solve for unknowns.

To solve the given problem, we rely on algebra to determine the years we are interested in (like 2008 and 2012), substituting these year values into the function. So, using algebra, we:

• Determine the time \(t\) since the base year 1992.
• Plug this \(t\) into the exponential equation to predict the population.

This is crucial as it helps us derive meaningful predictions about the future. By mastering algebra, one learns to decipher and interpret complex mathematical models with relative ease. This foundational skill allows us to engage in problem-solving in various academic and professional fields.