To determine the correct equation that represents Marco's savings scenario, we need to account for the following information:
- Marco's initial deposit, which is [tex]$8500.
- The annual interest rate, which is 7.25%.
The problem is about calculating the future value of the deposit using compound interest, which can be expressed using the compound interest formula:
\[A = P \left(1 + \frac{r}{n}\right)^{nt}\]
Where:
- \(A\) is the future value of the investment/loan, including interest
- \(P\) is the principal investment amount (the initial deposit or loan amount), which in this case is \$[/tex]8500
- [tex]\(r\)[/tex] is the annual interest rate (decimal), so 7.25% becomes 0.0725
- [tex]\(n\)[/tex] is the number of times that interest is compounded per year (assuming yearly compounding, [tex]\(n = 1\)[/tex])
- [tex]\(t\)[/tex] is the time the money is invested for in years, which is represented by [tex]\(x\)[/tex]
When interest is compounded once per year, the formula simplifies to:
[tex]\[A = P (1 + r)^t\][/tex]
Plugging in the given values:
- [tex]\(P = 8500\)[/tex]
- [tex]\(r = 0.0725\)[/tex]
- [tex]\(t = x\)[/tex] (years)
So, the equation becomes:
[tex]\[A = 8500 (1 + 0.0725)^x\][/tex]
Simplifying the expression inside the parentheses:
[tex]\[A = 8500 (1.0725)^x\][/tex]
Therefore, the equation that represents Marco's savings scenario is:
[tex]\[f(x) = 8500 \cdot 1.0725^x\][/tex]
So, the correct choice among the given options is:
[tex]\[f(x) = 8500 \cdot 1.0725^x\][/tex]
