Question 2 (Essay Worth 10 points)

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 of money in two accounts, A and B, over time.

The amount [tex]$f(x)$[/tex], in dollars, in account [tex]$A$[/tex] after [tex]$x$[/tex] years is represented by the function below:
[tex] f(x) = 9.628(0.92)^x [/tex]

Part A: Is the amount of money in account [tex]$A$[/tex] increasing or decreasing and by what percentage per year? Justify your answer. (5 points)

Part B: The table below shows the amount [tex]$g(r)$[/tex], in dollars, of money in account [tex]$B$[/tex] after [tex]$r$[/tex] years:
\begin{tabular}{|c|c|c|c|c|}
\hline [tex]$r$[/tex] (number of years) & 1 & 2 & 3 & 4 \\
\hline [tex]$g(r)$[/tex] (amount in dollars) & 8,972 & 8,074.80 & 7,267.32 & 6,540.59 \\
\hline
\end{tabular}

Which account recorded a greater percentage change in the amount of money over the previous year? Justify your answer. (5 points)



Answer :

### Part A

To determine whether the amount of money in account [tex]\( A \)[/tex] is increasing or decreasing, we analyze the function given:

[tex]\[ f(x) = 9.628(0.92)^x \][/tex]

Here, the base of the exponential function is [tex]\( 0.92 \)[/tex]. Since [tex]\( 0.92 \)[/tex] is less than 1, this indicates that the function is decreasing. In this context, each year the amount of money in account [tex]\( A \)[/tex] decreases.

To find the percentage decrease per year, we observe the base of the exponent:

[tex]\[ \text{Base} = 0.92 \][/tex]

The percentage decrease per year can be calculated as follows:

[tex]\[ \text{Percentage Decrease per Year} = (1 - \text{Base}) \times 100 \][/tex]
[tex]\[ \text{Percentage Decrease per Year} = (1 - 0.92) \times 100 \][/tex]
[tex]\[ \text{Percentage Decrease per Year} = 0.08 \times 100 \][/tex]
[tex]\[ \text{Percentage Decrease per Year} = 8\% \][/tex]

Thus, the amount of money in account [tex]\( A \)[/tex] is decreasing by 8% per year.

### Part B

The table shows the amount of money in account [tex]\( B \)[/tex] over different years:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline r \text{ (number of years)} & 1 & 2 & 3 & 4 \\ \hline g(r) \text{ (amount in dollars)} & 8,972 & 8,074.80 & 7,267.32 & 6,540.59 \\ \hline \end{array} \][/tex]

We will calculate the percentage change for each year in account [tex]\( B \)[/tex]:

From year 1 to year 2:
[tex]\[ \text{Percentage Change} = \left( \frac{8972 - 8074.80}{8972} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage Change} = \left( \frac{8972 - 8074.80}{8972} \right) \times 100 \approx 10\% \][/tex]

From year 2 to year 3:
[tex]\[ \text{Percentage Change} = \left( \frac{8074.80 - 7267.32}{8074.80} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage Change} = \left( \frac{8074.80 - 7267.32}{8074.80} \right) \times 100 \approx 10\% \][/tex]

From year 3 to year 4:
[tex]\[ \text{Percentage Change} = \left( \frac{7267.32 - 6540.59}{7267.32} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage Change} = \left( \frac{7267.32 - 6540.59}{7267.32} \right) \times 100 \approx 10\% \][/tex]

Here are the percentage changes for each year:
- Year 1 to 2: 10%
- Year 2 to 3: 10%
- Year 3 to 4: 10%

The greatest percentage change in the amount of money in account [tex]\( B \)[/tex] over the previous years is [tex]\( 10\% \)[/tex].

### Conclusion:
- In Part A, the amount of money in account [tex]\( A \)[/tex] is decreasing by 8% per year.
- In Part B, the account [tex]\( B \)[/tex] recorded a greater percentage change of 10% in the amount of money over the previous year.