Solution
Question A
- The formula representing the growth/decay rate of the function is given by:
[tex]\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}[/tex]
- Comparing this formula with the function given, we have:
[tex]\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}[/tex]
- 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:
[tex]\begin{gathered} \Delta=\frac{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}[/tex]
- We can simplify the formula further as follows:
[tex]\begin{gathered} \Delta=\frac{G(r+1)-G(r)}{(r+1)-r}=\frac{G(r+1)-G(r)}{r-r+1} \\ \\ \therefore\Delta=G(r+1)+G(r) \end{gathered}[/tex]
- Now, let us apply the formula to solve the question:
[tex]\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}[/tex]
- 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.
