Question

The amount of money in an account may increase due to rising stock prices and decrease due to falling stock prices. Marco is studying the change in the amount ofmoney in two accounts, A and B. over time.The amount f(x), in dollars, in account A after x years is represented by the function below:f(x) = 9,628(0.92)*Part A: Is the amount of money in account A increasing or decreasing and by what percentage per year? Justify your answer. (5 points)Part B: The table below shows the amount g(r), in dollars, of money in account B after r years:r (number of years)1231g(r) (amount in dollars) 8972 8 074 80 7 267 376 540 59AWhich account recorded a greater percentage change in amount of money over the previous vear? Justify vour answer. (5 points)

Answer 1

The Solution:

Given the function money in account A as below:

$f(x)=9628(0.92)^x\ldots\text{eqn}(1)$$\begin{gathered} f(x)=\text{amount of money (in dollars)} \\ x=\text{ number of years.} \end{gathered}$

Part A:

Comparing eqn(1) with the general formula below:

$f(x)=p(1-\frac{r}{100})^x$

We get

$\begin{gathered} p=9628\text{ dollars} \\ \end{gathered}$$\begin{gathered} 1-\frac{r}{100}=0.9628 \\ \\ 1-0.9628=\frac{r}{100} \\ \\ \frac{r}{100}=0.0372 \end{gathered}$

Cross multiplying, we get

$\begin{gathered} r=100\times0.0372 \\ r=3.72\text{\%} \end{gathered}$

Thus, the amount of money in account A is decreasing since 0.92 is less than 1.

It is decreasing at 3.72% per year.

Part B:

Given the table below:

To compare the rate of change of the amount of money in account A and account B.

We shall find the rate of change in the amount of money in account B.

By formula,

$\begin{gathered} \text{ Rate of change =}\frac{y_2-y_1}{x_2-x_1} \\ \text{Where} \\ x_1=1,y_1=8972 \\ x_2=2,y_2=8074.80 \end{gathered}$

Putting these values in the formula, we get

$\text{Rate of change =}\frac{8074.80-8972}{1-2}=\frac{-897.20}{-1}=897.20\text{ dollars}$

But for the money in account A, the rate of change per year is:

$\text{ Rate=}\frac{8972-8074.80}{8972}\times100$$\text{ rate =}\frac{897.20}{8972}\times100=0.1\times100=10\text{\%}$

Thus, account B recorded a greater percentage change in the amount of money than account A, since the rate of account B (10%) is greater than 3.72% of account A.