Question

The data in the following table show that the percentage of adults in the United States who are currently married is declining.

$$\begin{array}{|cc|}\hline \text{Year}& \text{Percent of Adults}\\ \text{Who Are Married}\\ 1960& 72.2\mathrm{\%}\\ 1980& 62.3\\ 2000& 57.4\\ 2010& 51.4\\ 2012& 50.5\\ \hline\end{array}$$
Assuming that the percentage of adults who are married will continue to decrease according to the exponential decay model:
a) Use the data for $1960and2012$ 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 $2015\text{and in}2018$.
c) At this decay rate, in which year will the percent of adults who are married be $40\mathrm{\%}?$

Answer 1

Step- 1: (a) Write an exponential function that describe the percent of adult married after time $t$ in years.

The equation of exponential growth or decay after time $t$ unit with initial quantity $A_0$at $t=0$ is $A=A_0{e}^{\pm kt}orA=A_0{e}^{-kt}$ respectively.

According to given data, the initial time is $1960$ whose corresponding data of percent of adult is $A_0=72.2$, so percent of adults after $t$ years is

$A_t=72.2e^{-kt}$

And the percent of adult in $2012$ can be written as $50.5=72.2e^{-kt}$where $t=2012-1960=52years$

Now, Find $k$ by solving equation $50.5=72.2e^{-k\left(52\right)}$:

$$\begin{gathered} \Rightarrow 50.5=72.2e^{-k\left(52\right)} \\ \Rightarrow \frac{50.5}{72.2}=e^{-k\left(52\right)} \\ \Rightarrow 0.70=e^{-k\left(52\right)} \\ \Rightarrow \ln \left(0.70\right)=-k\left(52\right) \\ \Rightarrow -0.36=-k\left(52\right) \\ \Rightarrow k=0.0069 \end{gathered}$$

Hence, the value of $k$ is $0.0069$ and exponential function that describes the percent of adults married after time $t$, in years is $A=72.2e^{-0.0069t}$.

Step- 2: (b) Estimate the percent of adults who are married by $2015a\text{nd in}2018$ using exponential function $A=72.2e^{-0.0069t}$.

The percent of adults who will be married in $2015$, so substitute $t=55$ years in $A=72.2e^{-0.0069t}$and simplify:

$\begin{array}{rcl}A& =& 72.2e^{-0.0069\left(55\right)}\\ & =& 72.2e^{-0.3795}\\ & =& 72.2\left(0.6842\right)\\ & \approx & 49.4\end{array}$

In the same manner, find the percent of adults who will be married by$2018$, so substitute $t=58$ years in $A=72.2e^{-0.0069t}$and simplify:

$\begin{array}{rcl}A& =& 72.2e^{-0.0069\left(58\right)}\\ & =& 72.2e^{-0.4002}\\ & =& 72.2\left(0.6702\right)\\ & \approx & 48.3\end{array}$

Hence the percent of adults who are married in $2015$ and in 2018 is $49.4\%$ and $48.3\%$ receptively.

Step 3: (c) Find the year when the percent of adults who are married be $40\%$.

Substitute, $A=40$ in $A=72.2e^{-0.0069t}$ and solve for $t$:

$\begin{array}{rcl}40& =& 72.2e^{-0.0069t}\\ & \Rightarrow & \frac{40}{72.2}=e^{-0.0069t}\\ & \Rightarrow & 0.5540=e^{-0.0069t}\\ & \Rightarrow & -0.5906=-0.0069t\\ & \Rightarrow & t=85.59\\ & \Rightarrow & t\approx 86\end{array}$

$\because$Initial year is $1960$, after $86$ years of time it will be $1960+86=2046$

Hence the year when the percent of adults who are married be $40\%$ is $2046$.

Hence,

a) The value of $k=0.0069$

b) The percent of adults who are married in $2015$ and in 2018 is $49.4\%$ and $48.3\%$ receptively.

c) The year when the percent of adults who are married be $40\%$ is $2046$.