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The data in the following table show that the percentage of adults in the United States who are currently married is declining.\begin{array}{|c|c|}\hline \text { Year } & \text { Percent of Adults } \\\\\text { Who Are Married } \\\1960 & 72.2 \% \\\1980 & 62.3 \\\2000 & 57.4 \\\2010 & 51.4 \\\2012 & 50.5 \\\\\hline\end{array} Assuming that the percentage of adults who are married will continue to decrease according to the exponential decay model: a) Use the data for 1960 and 2012 to find the value of \(k\) and to write an exponential function that describes the percent of adults married after time \(t\), in years, where \(t\) is the number of years after 1960 . b) Estimate the percent of adults who are married in 2015 and in 2018 c) At this decay rate, in which year will the percent of adults who are married be \(40 \% ?\)

Step by step solution

01

- Define the exponential decay model

The exponential decay model is given by the equation \[ P(t) = P_0 e^{kt} \]where \ P(t) is the percent of adults who are married at time t years after 1960, \ P_0 is the initial percent of adults who are married in 1960, \ k is the decay constant, and \ t is the number of years after 1960.

02

- Use the initial percentage

Identify the initial percentage of adults who are married in 1960: \[ P_0 = 72.2\text{\text{%}} \]

03

- Use the percentage in 2012

Identify the percentage of adults who are married in 2012: \[ P(52) = 50.5\text{\text{%}} \]Note that 2012 is 52 years after 1960.

04

- Solve for the decay constant k

Rearrange the exponential decay model to solve for k:\[ P(t) = P_0 e^{kt} \rightarrow 50.5 = 72.2 e^{52k}\]Divide both sides by 72.2:\[ \frac{50.5}{72.2} = e^{52k} \]Take the natural logarithm of both sides to solve for k:\[ \text{ln}\left( \frac{50.5}{72.2} \right) = 52k \rightarrow k = \frac{\text{ln}\left( \frac{50.5}{72.2} \right)}{52} \approx -0.0062 \]

05

- Write the exponential function

Substitute the values of \(P_0\) and \(k\) into the exponential decay model:\[ P(t) = 72.2 e^{-0.0062t} \]

06

- Estimate the percent of adults married in 2015

Calculate the percentage of adults who are married in 2015 (55 years after 1960):\[ P(55) = 72.2 e^{-0.0062 \times 55} \approx 49.4\text{\text{%}} \]

07

- Estimate the percent of adults married in 2018

Calculate the percentage of adults who are married in 2018 (58 years after 1960):\[ P(58) = 72.2 e^{-0.0062 \times 58} \approx 48.5\text{\text{%}} \]

08

- Find the year when the percentage will be 40%

Set \(P(t) = 40\%\):\[ 40 = 72.2 e^{-0.0062t} \]Solve for \(t\):\[ \frac{40}{72.2} = e^{-0.0062t} \rightarrow \text{ln}\left( \frac{40}{72.2} \right) = -0.0062t \rightarrow t = \frac{\text{ln}\left( \frac{40}{72.2} \right)}{-0.0062} \approx 94 \]So, the percentage will be 40% around the year 1960 + 94 = 2054.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Function

An exponential function is a mathematical representation often used to model growth or decay processes in various fields, such as biology, finance, and physics. It takes the form \[ P(t) = P_0 e^{kt} \], where \( P(t) \) represents the quantity at time \( t \), \( P_0 \) is the initial amount, \( e \) is the base of natural logarithms (approximately equal to 2.71828), and \( k \) is a constant that determines the rate of growth or decay.
For this exercise, we are modeling the percentage of adults who are married over time, recognizing a consistent decline. Thus, the 'decay' aspect of the exponential function is emphasized, giving us an exponential decay model.

Decay Constant

The decay constant, denoted as \( k \), plays a crucial role in an exponential decay function. It essentially determines how quickly the quantity decreases over time. In the given model, \( k \) is a negative value because it represents a decay or reduction.

To find \( k \), we used the data points for the years 1960 and 2012. We start with the equation \(\text{ln}\frac{P(t)}{P_0} = kt \), rearrange it to solve for \( k \), and substitute in the given percentages for those years. This calculation reveals that \( k \) is approximately -0.0062, indicating a slow but steady decline.

Natural Logarithm

The natural logarithm, often denoted as \( \text{ln} \), is the logarithm to the base \( e \) (where \( e \) is approximately 2.718). It is a powerful tool in various mathematical models, including exponential decay.

In our example, to isolate \( k \), we took the natural logarithm of both sides of the equation \( e^{52k} = \frac{50.5}{72.2} \). This allowed us to linearize the exponential term and solve for the decay constant \( k \). Utilizing natural logs simplifies handling the exponential form, making calculations more manageable.

Time-Dependent Percentages

When dealing with exponential decay models, it's often necessary to estimate values at specific time points. This involves substituting the desired time \( t \) into the function to find the respective percentage.

For instance, to find the percentage of married adults in 2015, we computed \( P(55) = 72.2 e^{-0.0062 \times 55} \). Similarly, for 2018, we used \( t = 58 \) to get \( P(58) = 72.2 e^{-0.0062 \times 58} \). These calculations help in projecting future trends based on current data.

Mathematical Modeling

Mathematical modeling involves creating equations and formulas to represent real-world situations. In our exercise, we used an exponential decay model to describe a societal trend—the reduction in the percentage of married adults over the years.
Mathematical modeling enables us to make predictions and informed decisions. By calculating when the percentage will reach a certain threshold, like 40%, we can predict future scenarios, in this case, around the year 2054. This approach highlights the practical application of mathematics in analyzing and forecasting social trends.