Question

The rate of increase of the number of married couples \(M\) (in thousands) in the United States from 1970 to 2005 can be modeled by \(\frac{d M}{d t}=1.218 t^{2}-44.72 t+709.1\) where \(t\) is the time in years, with \(t=0\) corresponding to \(1970 .\) The number of married couples in 2005 was 59,513 thousand. (Source: U.S. Census Bureau) (a) Find the model for the number of married couples in the United States. (b) Use the model to predict the number of married couples in the United States in \(2012 .\) Does your answer seem reasonable? Explain your reasoning.

Short answer

The model for the number of married couples in the U.S. and its prediction for the year 2012 will depend on the calculated constant of integration. The reasonableness of the prediction can be evaluated by comparing it with the 2005 value.

Step-by-step solution

Solving the Differential Equation

The given rate of increase of the number of couples is the derivative of the number of couples. Therefore, to get the actual number of married couples, which is the model asked, we need to integrate the given equation: \[M(t) = \int (1.218 t^{2}-44.72 t+709.1) dt = 0.406 t^3 - 22.36 t^2 + 709.1t + C,\] where \(C\) is the constant of integration.

Finding the constant of integration

We know that in 2005 (t=35), the number of married couples was 59,513. So we can use the model \(M(t)\) with \(t=35\) to get \(C\): \[59513 = 0.406(35)^3 - 22.36(35)^2 + 709.1(35) + C.\] Solving this equation will give the constant of integration.

Predict the number of married couples in 2012

Using the value of \(C\) from Step 2, we now have the model which can be used to predict the number of married couples in 2012. That is \(t=42\), since \(t=0\) corresponds to 1970. Substituting \(t=42\) into the model will yield the prediction.

Evaluate the answer

Checking the reasonableness of the calculated value against the given 2005 value should tell if the prediction is reasonable or not.

Key concepts

Integrating Factor

An integrating factor is a function used to simplify the process of solving certain types of differential equations, particularly linear first-order differential equations. It is essentially a function that, when multiplied by the differential equation, allows the equation to be written as the derivative of a product of two functions, making it solvable via integration.

While the concept of the integrating factor wasn't explicitly used in the exercise regarding the number of married couples, understanding it can be crucial when facing first-order differential equations that are not readily integrable. The integrating factor method provides a systematic way to make such equations manageable, and it is a key tool in the study of differential equations within mathematical modeling.

Initial Value Problem

An initial value problem in the context of differential equations is a problem where the solution to the differential equation must satisfy a specific initial condition. This condition is typically given in the form of the value of the unknown function at a certain point.

For example, in our exercise, we had to integrate the rate of increase to find the model for the number of married couples, but we needed one more piece of information to fully determine the function. Knowing the number of married couples in 2005 provided us with this initial condition. That value allowed us to calculate the constant of integration and thus correctly model the population of married couples.

Mathematical Modeling

The process of constructing mathematical models involves creating abstract representations (models) that can simulate real-world processes. It's a way to translate a situation into mathematical language, using equations to predict or describe phenomena.

The exercise we discussed serves as a prime example. The rate of increase of married couples was modeled with a second-order polynomial in time. By integrating this rate, we obtain a model that can be used for making predictions about the future or understanding past trends. Mathematical modeling is a powerful tool that crosses disciplines and allows us to make informed decisions based on quantitative data.

Population Growth Model

A population growth model is a mathematical framework used to represent the change in population size over time. There are many types of population models, from simple to complex. The most basic is an exponential growth model, but in real-life situations, more sophisticate models, like logistic growth, may be required.

In the context of our exercise, we used a polynomial model to represent the growth of married couples, which is a less common but still a valid method for describing population dynamics over a specific period. This model provided a more accurate representation of the actual changes over time and gave a platform to predict future population sizes given the historical data.