Problem 49 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The percentage of families with children that are headed by single females grew at the rate of $$ R(t)=0.8499 t^{2}-3.872 t+5 \quad(0 \leq t \leq 3) $$ households/decade between \(1970(t=0)\) and \(2000(t=3)\). The number of such households stood at \(5.6 \%\) of all families in 1970 . a. Find an expression giving the percentage of these households in the \(t\) h decade. b. If the trend continued, estimate the percentage of these households in 2010 . c. What was the net increase in the percentage of these households from 1970 to \(2000 ?\)

Step-by-Step Solution

Answer

In summary, the expression for the percentage of single female-headed households with children at time \(t\) is \(P(t) = 0.8499 \cdot \frac{1}{3}t^3 - 3.872 \cdot \frac{1}{2}t^2 + 5t + 5.6\). If the trend continued till 2010, the percentage would be \(11.56\%\). The net increase in the percentage of these households from 1970 to 2000 is \(3.34\%\).

Step 1: Find an expression for the percentage of households at time t

To find an expression for the percentage of households at time \(t\), we need to integrate the rate function \(R(t)\). Integration will give us a function representing the accumulated value of the rate up to time \(t\). Let's denote the function we're looking for by \(P(t)\), for the percentage of households. Then, we have: $$ P(t) = \int R(t) dt = \int (0.8499t^{2}-3.872t+5) dt $$ Now, we integrate each term separately: $$ P(t) = 0.8499 \int t^{2} dt - 3.872 \int t dt + 5 \int dt $$

Step 2: Integrate the rate function

We now perform the integration for each term: $$ P(t) = 0.8499 \cdot \frac{1}{3}t^3 - 3.872 \cdot \frac{1}{2}t^2 + 5t + C $$ Here, \(C\) is an integration constant. We can find its value using the initial condition that the percentage was \(5.6\%\) in 1970, i.e., when \(t=0\): $$ P(0) = 0.8499 \cdot \frac{1}{3}(0)^3 - 3.872 \cdot \frac{1}{2}(0)^2 + 5(0) + C = 5.6 $$ This simplifies to \(C = 5.6\). Therefore, the expression for the percentage of households at time \(t\) is: $$ P(t) = 0.8499 \cdot \frac{1}{3}t^3 - 3.872 \cdot \frac{1}{2}t^2 + 5t + 5.6 $$

Step 3: Estimate the percentage in 2010

To estimate the percentage of these households in 2010, we need to evaluate our expression for the time \(t=4\) (as 2010 is 4 decades after 1970): $$ P(4) = 0.8499 \cdot \frac{1}{3}(4)^3 - 3.872 \cdot \frac{1}{2}(4)^2 + 5(4) + 5.6 $$ Calculate the value: $$ P(4) \approx 11.56 $$ So, the estimate for the percentage of these households in 2010 is around \(11.56\%\).

Step 4: Calculate the net increase from 1970 to 2000

To find the net increase in the percentage of these households from 1970 to 2000, we need to calculate the difference between the values of our expression at \(t=0\) and \(t=3\): $$ \Delta P = P(3) - P(0) = \left[0.8499 \cdot \frac{1}{3}(3)^3 - 3.872 \cdot \frac{1}{2}(3)^2 + 5(3) + 5.6\right] - 5.6 $$ Calculate the value: $$ \Delta P \approx 3.34 $$ The net increase in the percentage of these households from 1970 to 2000 is around \(3.34\%\).

Key Concepts

Integration in Mathematics

Understanding the concept of integration in mathematics is crucial, especially when dealing with real-world applications such as predicting the percentage of households over time, as seen in the given exercise.

Integration is a fundamental part of calculus, closely related to differentiation, but while differentiation is about finding rates of change, integration is about accumulation. The process of integration, also known as anti-differentiation, sums up parts to find a whole. In the context of the exercise, integration helped to transform the rate of change given by the function R(t) into a cumulative percentage, P(t), over the decades.

Here are a few key points to remember about integration:

• Integration allows us to find the total accumulation of a quantity that changes over time.
• The integration constant C is determined using an initial condition or boundary value.
• The definite integral can be used to calculate the total increase or decrease over an interval, which corresponds to the net change in the exercise.

By mastering the technique of integration, we can solve many practical problems ranging from physics to finance and, as depicted here, in sociological statistics.

Sociological Statistics

Sociological statistics are invaluable when it comes to understanding changes and patterns in societies. The use of statistical methods helps sociologists to quantify behaviors, trends, and attributes of large groups of people, similar to what is seen in the exercise.

When working with sociological data, such as the percentage of single-female-headed households, it's important to:

• Understand that sociological statistics often deal with large population samples and diverse groups.
• Recognize that this statistical data can reflect important societal changes and help in formulating policies.
• Know that statistics need to be interpreted within the context of social theories and frameworks.

In the example, the statistical method of integration was used to predict the percentage of a certain type of household, demonstrating the power of sociological statistics in forecasting social trends based on past data.

Mathematical Modeling

Mathematical modeling is an insightful tool for capturing the complex nature of reality through the language of mathematics. In the given exercise, mathematical modeling is used to represent the rate of change in household dynamics.

A robust mathematical model goes through several phases:

• Formulation: Translating the real-world scenario into a mathematical framework.
• Solving: Applying mathematical methods to get solutions to the model.
• Interpretation: Converting the mathematical results back into real-world insights.

Through mathematical modeling, such as the polynomial function R(t), you can analyze and predict trends that inform better decision-making. In the context of the exercise, the model not only estimated the percentage of households at any given time but also uncovered the broader trend within the society from 1970 to 2000 and projected beyond.