Problem 38 - Calculus Concepts

Question

Marriage Age Between 1950 and 2000 , the rate of change in the rate at which the median age of first marriage of females in the United States was changing was constant at 0.0042 year of age per year squared. The median age of first marriage for these females was increasing at the rate of 0.1713 year of age per year in \(1991,\) and females were first married at a median age of 25.1 in 2000 . (Source: Based on data from www.infoplease.com) a. Write a differential equation for the rate at which the rate of the median age of first marriage for U.S. females is changing. b. Find a particular solution to the differential equation in \(\operatorname{part} a\) c. Use the result of part \(b\) to estimate the median age of first marriage of U.S. females in the current year.

Step-by-Step Solution

Answer

The differential equation is \(m''(t) = 0.0042\). The solution \(m(t)\) is determined by integrating and using given conditions. Substitute the current year into this function for an estimate.

Step 1: Understand the Given Information

We are given that the rate of change of the rate at which the median age of first marriage was changing is 0.0042 years of age per year squared. This refers to the second derivative, or acceleration, of the median age function. We also know the median age was increasing at the rate of 0.1713 year of age per year in 1991, and the median age was 25.1 years in 2000.

Step 2: Write the Differential Equation

The rate of change in the rate at which median age of marriage was changing is given by a constant rate. This means we have a constant second derivative. Let t represent the year and m(t) be the median age in year t. We have: \[ m''(t) = 0.0042.\]

Step 3: Integrate the Differential Equation

To find the general solution for the median age function, integrate the differential equation. First, integrate \(m''(t) = 0.0042\) to find \(m'(t):\) \[ m'(t) = 0.0042t + C,\] where \(C\) is a constant of integration.

Step 4: Use Initial Conditions to Determine First Constant

We know that in 1991, the rate of change was \(0.1713\). Setting \(t = 1991\), we solve:\(m'(1991) = 0.0042 \times 1991 + C = 0.1713.\) Solve for \(C\):\[ C = 0.1713 - 0.0042 \times 1991,\] which will give the exact value of \(C.\)

Step 5: Integrate Again to Find the General Solution

Integrate \(m'(t)\) to find \(m(t)\): \[ m(t) = \int 0.0042t + C \, dt = 0.0042 \frac{t^2}{2} + Ct + D,\] where \(D\) is another constant of integration.

Step 6: Use Known Median Age to Find the Second Constant

Now use the information that in 2000, \(m(2000) = 25.1\). Substitute into the equation to solve for \(D\):\[ m(2000) = 0.0021 \times 2000^2 + C \times 2000 + D = 25.1.\]Substitute the calculated value of \(C\) from Step 4 and solve for \(D\).

Step 7: Apply the Solution to Estimate Current Median Age

Now that you have the specific function \(m(t)\), substitute the current year into the equation to find the current median age of first marriage. For example, if the current year is 2023, calculate \(m(2023)\) using your equation.

Key Concepts

median age

The median age is a statistical measurement that represents the middle point in a dataset of ages. Specifically, it is the age where half of the population is younger, and the other half is older. In the context of first marriages in the U.S., the median age helps reveal how old individuals typically are when they first get married. Over time, this median age can shift due to various social and economic factors, such as changes in cultural norms, economic conditions, or education levels.

Understanding the median age is crucial because it provides insights into societal trends and expectations. For instance, a rising median age of first marriage could indicate a trend towards individuals prioritizing education or careers before settling down. This can have broad implications for other aspects of life like childbearing, household formation, and even economic markets focusing on different age demographics.

first marriage

The age at which individuals experience their first marriage can vary greatly based on cultural, economic, and personal factors. Historically, many factors contribute to the age people choose to marry for the first time, from educational pursuits to shifts in personal priorities, and societal expectations.

First marriages are significant life events marked by legal and often religious ceremonies that join two individuals together. The age when people first marry often reflects broader societal shifts. For example, delays in marriage age have been linked to increased participation in higher education and workforce, as well as a greater focus on personal growth and financial stability before joining in marriage.

Analyzing trends in the age of first marriages helps demographers and policymakers understand the relationship dynamics and family planning strategies within a country. It also provides insight into how societal roles and expectations may be evolving over time.

rate of change

In mathematical terms, the rate of change defines how one quantity alters as another configuration changes. When applied to the context of the median age of first marriages, the rate of change can help forecast future trends or identify historical patterns within the population.

For example, in the given problem, the rate of change in the age at first marriage was 0.1713 year of age per year in 1991 and the rate of change of this rate (second derivative) was a constant 0.0042 year of age per year squared. This kind of information allows us to create models that predict how the median age will change over time.

The concept of rate of change is often expressed and calculated using differential equations, which involve derivatives. A first derivative tells us the speed of change, while a second derivative (the rate of the rate of change) can show acceleration or deceleration in trends. Understanding these equations is crucial for predicting and planning around demographic shifts.