Question

The amount of money in an account may increase due to rising stock prices and decrease due to falling stock prices. Maggie is studying the change in the amount of money 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)xPart A: Is the amount of money in account A increasing or decreasing and by what percentage per year? Justify your answer. Part B: The table below shows the amount g(r), in dollars, of money in account B after r years.r (number of years)1234g(r) (amount in dollars)8,9728,074.807,267.326,540.59Which account recorded a greater percentage change in amount of money over the previous year? Justify your answer.

Answer 1

Question A

- The formula representing the growth/decay rate of the function is given by:

`\begin{gathered} f(x)=P(1+r)^x \\ \text{where,} \\ x=\text{Number of years} \\ r=\text{percentage increase/decrease per year.} \\ P=\text{The initial amount} \end{gathered}`

- Comparing this formula with the function given, we have:

`\begin{gathered} f(x)=P(1+r)^x \\ f(x)=9628(0.92)^x \\ \\ \therefore1+r=0.92 \\ \text{Subtract 1 from both sides} \\ r=0.92-1 \\ r=-0.08\equiv-8\text{ \%} \end{gathered}`

- The rate is a negative rate, thus, we can conclude that the amount in Account A is Decreasing and it's decreasing at 8% per year.

Question B

- The formula we will use to find the rate of change from year to year is:

`\begin{gathered} \Delta=(G(r+1)-G(r))/((r+1)-r) \\ \text{where,} \\ G(r+1)\text{ is the amount in the }(r+1)^(th)\text{ year.} \\ G(r)\text{ is the amount in the }r^(th)\text{ year} \\ (r+1)\text{ is the next year} \\ r\text{ is the current year} \end{gathered}`

- We can simplify the formula further as follows:

`\begin{gathered} \Delta=(G(r+1)-G(r))/((r+1)-r)=(G(r+1)-G(r))/(r-r+1) \\ \\ \therefore\Delta=G(r+1)+G(r) \end{gathered}`

- Now, let us apply the formula to solve the question:

`\begin{gathered} \Delta_(2-1)=8074.80-8972=-897.20 \\ \Delta_(3-2)=7267.32-8074.80=-807.48 \\ \Delta_(4-3)=6540.59-7267.32=-726.73 \\ \\ \Delta_(3-2)\text{ is the greatest change from YEAR 2 to YEAR 3} \end{gathered}`

- The question we are asked to solve for Question B is vague. I cannot proceed from here.

Final Answer

Question A

The rate is a negative rate, thus, we can conclude that the amount in Account A is Decreasing and it's decreasing at 8% per year.