Problem 30 - Precalculus : Building Concepts and Connections

Question

The population of the United States is expected to grow from 282 million in 2000 to 335 million in \(2020 .\) (Source: U.S. Census Bureau) (a) Find a function of the form \(P(t)=C e^{k t}\) that models the population growth of the United States. Here, \(t\) is the number of years since 2000 and \(P(t)\) is in millions. (b) Assuming the trend in part (a) continues, in what year will the population of the United States be 300 million?

Step-by-Step Solution

Answer

The function that models the population growth is \(P(t) = 282 e^{k t}\) where \(k = \ln(\frac{335}{282}) / 20\). The population of the United States will reach 300 million in the year \(2000 + \frac{\ln(\frac{300}{282})}{k}\).

Step 1: Calculate the constant \(C\)

In the year 2000, the population was 282 million and since \(t\) is calculated from the year 2000, \(P(0) = 282\). Substituting this into the equation \(P(t)=C e^{k t}\) gives the constant \(C\), so \(C = 282\). Therefore, the equation becomes \(P(t)=282 e^{k t}\).

Step 2: Find the growth constant \(k\)

In the year 2020, the population is expected to be 335 million. This corresponds to \(t=20\). Substituting the values \(P(20)=335\) and \(C=282\) into the equation gives \(335 = 282 e^{20k}\). Solving for \(k\) requires rearranging the equation as \(k = \ln(\frac{335}{282}) / 20\). This provides the value of \(k\). Now, all parameters of the model are known.

Step 3: Find the year when population will be 300 million

To find the year when the population will be 300 million, we need to solve for \(t\) when \(P(t) = 300\). This is achieved by rearranging the equation to look like this: \(t = \frac{\ln(\frac{300}{282})}{k}\). Calculating this will give the value of \(t\) in years since 2000, which is the year when the population will reach 300 million.

Key Concepts

Population Modeling

Population modeling is a powerful tool used to study how populations change over time. It helps us predict future population sizes, which is crucial for planning resources, infrastructure, and policies. This form of modeling often uses mathematical functions to represent the growth or decline of a population. In this exercise, we're looking at the population of the United States using an exponential growth model.

Key elements in population modeling include:

• Initial Population: The starting size of the population, which in this case is 282 million in the year 2000.
• Time Variable: This represents the passage of time, often expressed in years from a specific starting point.
• Growth Patterns: Can be linear, exponential, logistic, etc., and they describe how populations increase or decrease.

Population modeling is essential for understanding demographic changes that impact environmental, social, and economic factors. By utilizing models, we gain valuable insights into potential future scenarios, helping organizations and governments make informed decisions.

Exponential Function

An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. In the context of population growth, the exponential function is particularly suited for modeling because many populations grow at rates proportional to their size.

We express the population growth function as \(P(t) = C e^{kt}\), where:

• \(C\) is the initial population size.
• \(e\) is the base of natural logarithms, approximately equal to 2.71828.
• \(k\) is the growth constant.
• \(t\) represents the time in years since the base year.

Exponential functions are characterized by rapid growth. Even small changes in the exponent, \(k\), can lead to significant increases in population size over time. Understanding how to manipulate and solve exponential functions is critical for accurate population predictions and assessments.

Growth Constant

The growth constant \(k\) in population modeling is a measure of how quickly a population is increasing or decreasing. It's a crucial part of the exponential model equation \(P(t) = C e^{kt}\) and determines the rate of change relative to the population size.

To find \(k\), we need data from at least two different points in time with known populations. In this problem, we have:

• The population in 2000: 282 million.
• The population in 2020: 335 million.

These values allow us to write the equation \(335 = 282 e^{20k}\). Solving for \(k\) involves taking the natural logarithm of both sides and rearranging the equation: \[k = \frac{\ln \left( \frac{335}{282} \right)}{20}\].

Once \(k\) is determined, it can be used to predict future population sizes or estimate when the population will reach a certain size. The growth constant is integral to managing and planning for the effects of population changes.