Problem 50 - Applied Calculus

Question

The divorce rate in the United States (divorces per year) has been declining in recent years. The number of divorces per year is predicted to be \(0.94 e^{-0.02 t}\) million, where \(t\) is the number of years since 2014 . a. Find a formula for the total number of divorces within \(t\) years of 2014 b. Use your formula to find the total number of divorces from 2014 to \(2020 .\) Source: National Center for Health Statistics

Step-by-Step Solution

Answer

The total number of divorces from 2014 to 2020 is approximately 5.31 million.

Step 1: Understand the Problem

We are given that the divorce rate in the U.S. is a function of time: \( R(t) = 0.94 e^{-0.02t} \), where \( t \) is the number of years since 2014. We need to find a formula for the cumulative number of divorces over \( t \) years and use that formula to calculate the total number of divorces from 2014 to 2020.

Step 2: Formulate the Function for Total Divorces

To find the total number of divorces within \( t \) years, we need to integrate the rate function \( R(t) \) from 0 to \( t \). This will give us the cumulative divorces, \( D(t) \). The integral is given by the formula: \[ D(t) = \int_{0}^{t} 0.94 e^{-0.02u} \, du \].

Step 3: Calculate the Indefinite Integral

First, find the indefinite integral of \( 0.94 e^{-0.02u} \) with respect to \( u \). This is computed as: \[ \int 0.94 e^{-0.02u} \, du = \frac{0.94}{-0.02} e^{-0.02u} = -47 e^{-0.02u} + C \], where \( C \) is the constant of integration.

Step 4: Evaluate the Definite Integral

Substitute the limits of integration into the indefinite integral to compute the total number of divorces within \( t \) years: \[ D(t) = \left[-47 e^{-0.02u}\right]_0^t = -47 e^{-0.02t} + 47 e^0 = 47(1 - e^{-0.02t}) \].

Step 5: Calculate Total Divorces from 2014 to 2020

To find the total divorces from 2014 to 2020, substitute \( t = 6 \) (since 2020 is 6 years after 2014) into \( D(t) \): \[ D(6) = 47(1 - e^{-0.02 \times 6}) \].

Step 6: Compute the Exact Value

Calculate \( e^{-0.12} \) and substitute to find \( D(6) = 47(1 - e^{-0.12}) \). Using a calculator, \( e^{-0.12} \approx 0.8869 \), thus \( D(6) \approx 47(1 - 0.8869) \approx 47 \times 0.1131 \approx 5.3147 \approx 5.31 \) million.

Key Concepts

Integration

Integration is a fundamental concept in calculus, often used to find the area under a curve. It serves as an essential tool for accumulating quantities, such as distance, area, volume, and in this case, the number of divorces over time. In our problem, we have a rate function, \( R(t) = 0.94 e^{-0.02t} \), representing the divorce rate at a given time.To find the total number of divorces over a period of \( t \) years, we integrate this rate function. The integration form is: \[ D(t) = \int_{0}^{t} 0.94 e^{-0.02u} \, du \]. This expression sums up the rate over time to provide the cumulative total. By integrating from 0 to \( t \), we effectively collect all instances of divorces happening over those \( t \) years. The result of this integration is the function \( D(t) = 47(1 - e^{-0.02t}) \), which gives us the total number of divorces up to that point. Performing definite integration helps translate a continuous rate into an accumulated value, such as total divorces. This process can similarly apply to various real-world situations, like total sales or cumulative growth, highlighting the power of integration in practical applications.

Exponential Functions

Exponential functions represent scenarios where change grows or decays at a consistent rate. They take the form \( f(t) = ae^{bt} \), where \( a \) scales the function and \( b \) determines the growth or decay rate.In our problem, the function \( R(t) = 0.94 e^{-0.02t} \) is an exponential decay function. The multiplier, \( 0.94 \), indicates the initial divorce rate when \( t = 0 \) (the year 2014). The exponent, \( -0.02t \), shows a continuous decline in the divorce rate over time, typical of `exponential decay`.A key characteristic of these functions is their rapid change, initially steep and gradually level off, reflecting minimal changes as \( t \) increases. This behavior is visually observable in curve graphs, appearing as a swooping drop-off for decays.Exponential functions are pivotal in modeling processes where outcomes either compactly reduce or amplify, like population dynamics, radioactive decay, or financial interest levied over time. Highlighting the divorce rate, it showcases how such formulas can project trends or future predictions.

Real-world Mathematical Modeling

Mathematical modeling uses equations to describe and predict real-world occurrences. It helps us interpret and foretell changes using measurable data, bringing abstract math to a tangible reality.In this exercise, the model provides insights into the social trend of divorce rates using the equation \( R(t) = 0.94 e^{-0.02t} \). By applying the model, we can estimate the impact and future trajectory of divorces over years starting from 2014.This model simplifies a complex human phenomenon into manageable numbers allowing decisions to be made based on predictions: - Potential policies could be influenced by decreasing rates.- Predictions serve demographers and sociologists to analyze societal changes. - Further refinements or updates to models can include more variables, reflecting further factors affecting divorce like employment or economic conditions.Overall, mathematical models bridge the gap between theory and practice, converting raw data into understandable formats guiding various fields of research and everyday decision-making.