Question

Population of West Virginia The population of West Virginia decreased from about 1.853 million in 2010 to 1.831 million in 2016\. Use an exponential model to predict the population in 2025. Explain why an exponential (decay) model might not be an appropriate long-term model of the population of West Virginia.

Short answer

Answer: According to the exponential decay model, West Virginia's population in 2025 is estimated to be around 1,801,000. This model might not be appropriate for long-term predictions because it assumes that factors influencing population changes, such as economic development, migration, and changes in birth and death rates, remain constant over time, which is often not the case.

Step-by-step solution

1. Establish the exponential decay model

To create an exponential decay model, we will use the formula P(t) = P0 * e^(kt), where P(t) is the population at time t, P0 is the initial population (in 2010), k is the decay constant, and t is the time in years since 2010. With the given data, we have P0 = 1,853,000 and P(6) = 1,831,000. We will substitute these values into the equation and solve for the decay constant k. 1,831,000 = 1,853,000 * e^(6k)

2. Calculate the decay constant (k)

Divide both sides by 1,853,000: e^(6k) = 1,831,000 / 1,853,000 ≈0.9881 Take the natural logarithm of both sides: 6k = ln(0.9881) Now, solve for k: k = ln(0.9881) / 6 ≈ -0.002

3. Use the exponential decay model to predict the population in 2025

We will now use the decay constant k and the exponential decay model to predict West Virginia's population in 2025. We need to calculate P(15) since 2025 is 15 years after 2010. P(15) = 1,853,000 * e^(-0.002 * 15) P(15) ≈ 1,853,000 * e^(-0.03) ≈ 1,801,000

4. Discuss the reliability of the exponential decay model

Although the exponential decay model can be a useful tool for predicting population changes in the short term, it may not be an appropriate model for long-term predictions. Population changes are often influenced by various factors such as economic development, migration, and changes in birth and death rates. Using an exponential decay model assumes that these factors remain constant over time, which is often not the case. For the population of West Virginia, factors such as the decline in the coal mining industry, changes in economic opportunities, and shifts in migration could influence population trends in unpredictable ways. Therefore, it is essential to consider other factors when making long-term predictions about population change to obtain more accurate results. In conclusion, according to the exponential decay model, West Virginia's population in 2025 is estimated to be around 1,801,000. However, it is essential to note that this model might not be entirely appropriate for long-term predictions due to the dynamic factors influencing population changes.

Key concepts

Population Prediction

Predicting a population using an exponential model involves estimating how a number of individuals within a defined geographical boundary may change over time. In this scenario, the population of West Virginia was examined utilizing an exponential decay formula. This method helps in understanding trends and patterns in population changes by modeling decreases over time.

The exponential decay model used, denoted by the formula $P(t)=P_0\cdot e^{kt}$, requires three main components:

• $P(t)$: the population at time $t$
• $P_0$: the initial population size
• $k$: the decay constant

In practice, such models assist in providing short-term predictions. However, they rely on the assumption that the decay rate remains constant, which is an important consideration when using the model in real-world applications.

Decay Constant

The decay constant, $k$, is a critical piece of the exponential decay model equation. It serves as a measure of the rate of change over time, reflecting how quickly or slowly the population decreases. Calculating $k$ involves using existing population data at two different points in time.

In the given problem, we have data from 2010 ($P_0=1,853,000$) and 2016 ($P(6)=1,831,000$). By substituting these values into the model $1,831,000=1,853,000\cdot e^{6k}$, and subsequently solving for $k$, we determine $k\approx -0.002$. This small negative value indicates a gentle decline over the 6-year period observed.

The decay constant is vital, as it effectively encapsulates the rate of population change due to factors such as emigration or declining birth rates, assuming other factors remain unchanged.

Exponential Function

The exponential function involved here, $e^{kt}$, is an essential part of exponential modeling. This mathematical operation allows for predictions of exponential growth or decay processes. The base $e\approx 2.71828$ is a fundamental mathematical constant used throughout various scientific disciplines.

In this context, the exponential function determines how the original population size, $P_0$, changes over a specific time interval, $t$. By applying this function with either a positive or negative $k$, we can model growth or decay, respectively. A negative $k$ as in the West Virginia example signals a decrease in population over time.

The strengths of the exponential function in modeling population or related phenomena lie in its capacity to simplify the complexity of natural decay processes into a manageable prediction. However, its practical application must also account for real-world variability.

Long-term Population Prediction Limitations

While the exponential decay model provides a useful tool for short-term predictions, its utility in long-term scenarios comes with limitations. Several factors can affect population dynamics over extended periods, which the model may not reflect accurately.

Factors affecting population include:

• Economic changes
• Migration trends
• Health advancements
• Environmental changes

The assumption that these factors remain constant often fails, especially in regions experiencing significant economic, political, or environmental shifts. For West Virginia, the decline in industries like coal mining, along with economic migration, could significantly alter population trends, posing challenges to the exponential model's predictions.

A thorough understanding of these limitations stresses the importance of complementing the exponential approach with additional data and growth models when devising long-term predictions.