Problem 88 - Applied Calculus

Question

SOCIOLOGY: Marriages The marriage rate (marriages per year) in the United States has been declining recently, with about \(1.97 e^{-0.0102 t}\) million marriages per year, where \(t\) is the number of years since 2014 . Assuming that this rate continues, find the total number of marriages in the United States from 2014 to 2024

Step-by-Step Solution

Answer

The total number of marriages in the US from 2014 to 2024 is approximately 18.76 million.

Step 1: Determine the initial information

We are given the marriage rate function as a function of time since 2014. The function is \( 1.97 e^{-0.0102 t} \) million marriages per year. We are tasked with finding the total number of marriages from 2014 to 2024, so \( t \) will range from 0 to 10.

Step 2: Set up the integral

To find the total number of marriages over a period of time, we need to integrate the rate function with respect to time. The integral we need to solve is: \[ \int_{0}^{10} 1.97 e^{-0.0102 t} \, dt \]. This will give us the total number of marriages in million from 2014 to 2024.

Step 3: Compute the integral

Solve the integral \( \int 1.97 e^{-0.0102 t} \, dt \), which yields \(-\frac{1.97}{0.0102} e^{-0.0102 t} + C\). Evaluating it from 0 to 10 gives us: \[\left[-\frac{1.97}{0.0102} e^{-0.0102 \cdot 10}\right] - \left[-\frac{1.97}{0.0102} e^{-0.0102 \cdot 0}\right] = \frac{1.97}{0.0102} (1 - e^{-0.102})\].

Step 4: Simplify the expression

Calculate \( \frac{1.97}{0.0102} \), which is approximately 193.14, and \( 1 - e^{-0.102} \), which is approximately 0.0971. The total number of marriages in millions is then \[ 193.14 \times 0.0971 \approx 18.76 \].

Step 5: Interpret the result

Multiply 18.76 million by 1 million to put the number into perspective as 18760000 marriages, which is the total number of marriages in the US from 2014 to 2024.

Key Concepts

Integral Calculus

Integral calculus is a branch of mathematics concerned with finding functions that describe accumulated quantities over a given interval. In the context of the original exercise, we use integral calculus to determine the total number of marriages over a period of years.
Integrals help us calculate areas under curves and, in practical scenarios like this one, can represent total quantities such as population, growth, or, in this case, marriages.

• To solve the given problem, we take the provided rate function, which tells us the marriage rate each year, and integrate it over the desired interval, specifically from 2014 to 2024.
• The integral set up is \[ \int_{0}^{10} 1.97 e^{-0.0102 t} \, dt \] which represents an accumulation of the marriage rate over 10 years.

In essence, integral calculus allows us to go from knowing a rate of change to knowing a total quantity over a period, providing a deeper understanding of cumulative processes in real-world situations.

Exponential Functions

Exponential functions are mathematical functions of the form \( a \cdot e^{bx} \), where \( e \) is the base of the natural logarithm, approximately equal to 2.71828, and \( a \) and \( b \) are constants. These functions are significant because they model processes that grow or decay at a rate proportional to their current value.

In the exercise, the marriage rate is expressed as an exponential function \( 1.97 e^{-0.0102 t} \). This particular form indicates an exponential decay due to the negative sign before the exponent.

• The function shows how the number of marriages decreases over time since 2014.
• Exponential functions are used in a variety of fields for modeling real-world scenarios where change is not linear but rather multiplicative.

Understanding exponential functions can help in predicting future trends and making informed decisions based on models of growth or decay.

Mathematical Modeling

Mathematical modeling involves using mathematics to represent, analyze, and/or predict real-world phenomena. This is crucial in fields such as economics, biology, and sociology. The marriage rate problem is an example of mathematical modeling, where we use a mathematical function to describe the yearly number of marriages over time.

The process involves:

• Identifying the real-world problem or scenario.
• Choosing appropriate mathematical frameworks (like exponential functions) for modeling.
• Deriving conclusions or making predictions based on the mathematical analysis.

In this scenario, mathematical modeling allows us to forecast the number of marriages over a decade based on an existing trend. Such models help sociologists, policymakers, and researchers understand patterns and devise strategies accordingly.

Sociology and Mathematics

Sociology and mathematics might seem like unrelated fields at first glance, but they often intersect through the application of mathematical methods in analyzing social patterns and trends. Mathematics provides tools for a systematic, quantitative understanding of social phenomena.

In the context of marriage rates, using a mathematical model helps sociologists track trends and possibly forecast future changes. This intersection allows:

• Collecting data and understanding shifts in social behaviors over time.
• Applying statistical and mathematical models to analyze societal issues.
• Creating foundational knowledge to support public policy and social planning.

Thus, through mathematical analysis, we can understand how shifts in marriage rates might impact other social and economic factors, testing various hypotheses with both historical data and predictive models.