Chapter 8, Problem 15STP

(a)

To determine

To find: The exponential function to predict the amount of money.

Answer to Problem 15STP

The required model is $Y=a{e}^{kt}$ .

Explanation of Solution

Given:

The amount inherited is $\$250,000$ .

The year of money inheritance is $1992$ .

The given table is shown in Table 1

Year	Amount
$1992$	$\$250,000$
$2000$	$\$329,202$
$2005$	$\$390,989$

Table 1

Calculation:

Consider the exponential function to predict the amount of money.

  $Y=a{e}^{kt}$

Here, $Y$ is the money after $t$ years, $a$ represents initial money $k$ rate at which money is growing per year.

(b)

To determine

To find: The year in which the money grows to amount $\$500,000$ .

Answer to Problem 15STP

The money increases to $\$500,000$ after 20 years that is in the year $2018$ .

Explanation of Solution

Given:

The final amount is $\$500,000$ .

Calculation:

Consider the formula for the number of money is,

  $Y=a{e}^{kt}$

Consider that in the year 2006 after the 8 years the money is increased to $\$329202$ . Substitute the values in the formula as,

  $\begin{array}{l}\$329202=\$250,000e^{k8}\\ \frac{\$329202}{\$250,000}=e^{k8}\\ \ln e^{k8}=\ln 1.317\\ k=\frac{0.257}{8}\end{array}$

Solve further as,

  $k=0.0314$

Consider the amount increases to $\$500,000$ in the year.

  $\begin{array}{l}\$500,000=\$250,000e^{8t}\\ \frac{\$500,000}{\$250,000}=e^{k8}\\ \ln e^{t8}=\ln 2\\ t=\frac{0.69}{0.0334}\end{array}$

Solve further as,

  $\begin{array}{c}t=\frac{0.69}{0.0334}\\ =20.05\end{array}$

The money increases to $\$500,000$ after 20 years that is in the year $2018$ .