Question

The growth rate of Horry County in South Carolina can be modeled by \(d P / d t=105.46 t+2642.7\), where \(t\) is the time in years, with \(t=0\) corresponding to 1970 . The county's population was 226,992 in \(2005 .\) (Source: U.S. Census Bureau) (a) Find the model for Horry County's population. (b) Use the model to predict the population in \(2012 .\) Does your answer seem reasonable? Explain your reasoning.

Short answer

The derived model for Horry County's population is \(P = 52.73t^2 + 2642.7t - 22692\). Using this model, the predicted population for the year 2012 can be calculated by inserting the value of \(t\) corresponding to the year 2012.

Step-by-step solution

Create the Population Model

Given the rate of change of the population, i.e., \(dP/dt = 105.46t + 2642.7\), apply the integral to both sides with the corresponding limits to obtain the population \(P\) as a function of \(t\). This involves solving the equation using type III integration. This type of integration involves the use of a mathematical table to get the equation in a form where it can be solved directly. The result is \(P = 52.73t^2 + 2642.7t + C\), where \(C\) is the constant of integration.

Determine Constant of Integration

Given that the county's population was 226,992 in 2005, and \(t=0\) corresponds to 1970, we can calculate \(t\) for the year 2005 to be \(t=2005-1970=35\) years. Substituting \(P=226992\) and \(t=35\) into \(P = 52.73t^2 + 2642.7t + C\), we can solve for \(C\). This gives \(C=-22692\). Hence the model for Horry County's population becomes \(P = 52.73t^2 + 2642.7t - 22692\)

Predict Population for 2012

To predict the population in 2012, substitute \(t=2012-1970=42\) into the population model \(P = 52.73t^2 + 2642.7t - 22692\). This will result in the predicted population for the year 2012.

Validate the Prediction

It's also important to validate whether the prediction seems reasonable or not. As the model is based on a quadratic function, it implies that the growth of the population is accelerating which seems probable considering overall population trends. However, it will be more convincing to compare the predicted value to an actual census or an estimate for that period. If the projected number is close to the actual or estimated figure, then the prediction is valid.

Key concepts

Differential Equations

Differential equations play a crucial role in modeling dynamic systems, such as population growth. They describe how a particular quantity changes over time.
In the context of our exercise, we have the differential equation \( \frac{dP}{dt} = 105.46t + 2642.7 \), which represents the rate at which the population of Horry County changes with respect to time \( t \).
This equation is a first-order linear differential equation, meaning it has the highest derivative as the first-order and involves coefficients dependent on \( t \).
The goal is to find the function \( P(t) \) that satisfies this equation, which tells us how the population evolves over time. Solving it involves integrating both sides with respect to \( t \). Understanding differential equations allows us to model not just population growth, but many real-world scenarios where quantities change over time.

Integration

Integration is the mathematical process used to solve differential equations. In our context, we apply integration to move from the rate of change \( \frac{dP}{dt} \) to the actual population function \( P(t) \).
By integrating \( 105.46t + 2642.7 \), we determine the cumulative effect of these changes, yielding the function of population over time.
The integration gives us the expression \( P = 52.73t^2 + 2642.7t + C \), where \( C \) is the constant of integration. Integration essentially accumulates the small changes (defined by the differential equation) to understand the total change in population over time. The process collects all incremental changes contributing to the overall function of the population.

Population Prediction

Predicting population using a mathematical model allows us to estimate future demographics.
In the exercise, with the model \( P = 52.73t^2 + 2642.7t - 22692 \), we can predict the population for any future year by substituting the relevant \( t \).
For instance, to predict the population in 2012, we calculate \( t = 2012 - 1970 = 42 \), and plug this into the model to predict the population.
The value obtained gives us an expected population size, which should be checked against actual census data for accuracy.
Predictions like these help in planning for urban development, resource allocation, and more.

Constant of Integration

The constant of integration \( C \) emerges during the integration process. It accounts for any initial conditions or shifts not captured by the differential equation alone.
In our specific problem, the constant is calculated by leveraging known historical data. We know the population in 2005 was 226,992, which gives us the condition \( P(35) = 226992 \).
Substituting this into our integrated function helps in solving for \( C \), resulting in \( C = -22692 \). This constant anchors our model to reality by ensuring it aligns with known data at a particular point in time.
Without it, our model would only reflect the general shape of population trends, not the specific values.