Problem 25 - Calculus Early Transcendentals

Question

Devise an exponential decay function that fits the following data; then answer the accompanying questions. Be sure to identify the reference point \((t=0)\) and units of time. The population of Michigan decreased from 9.94 million in 2000 to 9.88 million in 2010. Use an exponential model to predect the population in \(2020 .\) Explain why an exponential (decay) model might not be an appropriate long-term model of the population of Michigan.

Step-by-Step Solution

Answer

Answer: The predicted population of Michigan in 2020 using the exponential decay model is approximately 9.82 million. The limitations of using this model for long-term predictions include its assumption of a continuous decline in population at a constant rate, while ignoring real-world factors such as economic conditions, migration patterns, births, deaths, and government policies that can influence population growth or decline over time. This may cause the model to be less accurate for long-term projections.

Step 1: Identify the reference point and units of time

The reference point \((t=0)\) in this problem is the year 2000, which corresponds to the population value of 9.94 million. The units of time here will be in years.

Step 2: Create an exponential decay function

We will define the population, \(P(t)\), as a function of time, where \(P(0)=9.94\). An exponential decay function can be written in the form: \(P(t) = P_0\cdot e^{kt}\), where \(P_0\) is the initial population, \(k\) is the rate of decay, and \(t\) is time in years. We know that \(P_0=9.94\) million, and \(P(10)=9.88\) million. Using the exponential decay formula, we can set up an equation to solve for \(k\). Equation: \(9.88 = 9.94 \cdot e^{10k}\)

Step 3: Solve for the decay rate, k

Solve the equation for \(k\): \begin{equation} \frac{9.88}{9.94} = e^{10k} \end{equation} Taking the natural logarithm of both sides, \begin{equation} \ln\left(\frac{9.88}{9.94}\right) = 10k \end{equation} Now, divide by 10 to isolate \(k\): \begin{equation} k = \frac{1}{10} \ln\left(\frac{9.88}{9.94}\right) \end{equation} Calculating the value of \(k\), we get \(k \approx -0.0006026\)

Step 4: Write the exponential decay function

Plugging the value of \(k\) in the exponential decay function, we get: \begin{equation} P(t) = 9.94 \cdot e^{-0.0006026t} \end{equation}

Step 5: Predict the population in 2020

Year 2020 corresponds to \(t=20\). We plug in the value of \(t\) into the population function: \begin{equation} P(20) = 9.94 \cdot e^{-0.0006026 \cdot 20} \end{equation} Calculating the value, we get \(P(20) \approx 9.82\) million. The predicted population of Michigan in 2020 is approximately 9.82 million.

Step 6: Explain the limitation of the exponential decay model

An exponential decay model assumes a continuous decline in population at a constant rate. However, real-world factors such as economic conditions, migration patterns, births, deaths, and government policies can all influence population growth or decline over time. These factors can cause the rate of decay to change, making the exponential decay model less accurate for long-term projections. In reality, it is possible that the population of Michigan may stabilize or even increase at some point, rather than declining indefinitely as the exponential decay model suggests.

Key Concepts

Population Dynamics

Population dynamics is the study of how and why the characteristics of populations—such as size and age distribution—change over time. It involves understanding the birth rates, death rates, immigration, and emigration which affect population sizes. For any given region, such as Michigan, factors influencing these dynamics include:

• Economic conditions: Economic growth or recession can affect migration and birth rates.
• Natural resources: Availability and conservation of resources can impact population sustainability.
• Policies: Governmental policies on health, education, and immigration can significantly shift population trends.

Grasping population dynamics helps in making projections about the future population sizes, informing public policy, and aiding in resource management. However, these predictions can be complex due to the many variables involved.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent, represented as: \[ f(t) = a \cdot b^t \]This type of function is often used to model growth or decay processes where changes occur at a constant percentage rate over regular intervals. In exponential decay, like in the Michigan population example, the function lowers in value over time.

• The base of the exponent is less than one in decay scenarios, indicating a percentage decrease.
• These functions can efficiently capture rapid changes over time, making them useful in fields such as biology and economics.
• They provide a simple way to model complex systems provided the rate of change stays constant.

However, while useful, exponential functions may not always carpetbag all variables in real-world scenarios, which can hinder their accuracy in long-term predictions.

Mathematical Modeling

Mathematical modeling involves creating equations and computations to represent and analyze real-world systems. It provides a framework for understanding processes or predicting future outcomes by interpreting empirical data using mathematical expressions.

• Models like exponential decay simplify complex phenomena into understandable mathematical forms.
• They are highly effective in scenarios where certain conditions (such as constant rate of change) are met.
• The models can help visualize potential scenarios, offering insights and aiding decision-making.

The limitations, however, arise when models oversimplify. Especially in population studies, factors like unexpected migration or policy changes can undermine a model’s reliability, necessitating a careful evaluation and potentially multi-layered approaches for realistic predictions.

Decay Rate

The decay rate in exponential decay functions quantifies how quickly a population is decreasing over time. It is a critical component in understanding how the population size changes, expressed as a constant negative rate in the function:
\[ P(t) = P_0 \cdot e^{kt} \]where \( k \lt 0 \) represents the decay rate.

• In the case of Michigan's population, the decay rate \( k \approx -0.0006026 \) signifies a small, continuous yearly decline.
• Decay rates need precise calculation from accurate data points, which are crucial for modeling predictions.
• Even slight alterations in decay rate can substantially affect long-term forecasts.

Understanding decay rates helps in adjusting models to reflect real-world complexities, offering more robust views of potential future population shifts.