Answered

13. Marco is saving for tuition costs at a state university. He deposited $8500 in a 4-year certificate of deposit that earns 7.25% per year. What equation represents this scenario?

A. [tex]\( f(x) = 4 \cdot 7.25^x \)[/tex]
B. [tex]\( f(x) = 8500 \cdot 1.0725^x \)[/tex]
C. [tex]\( f(x) = 8500 \cdot 4^x \)[/tex]
D. [tex]\( f(x) = 1.0725 \cdot 8500^x \)[/tex]



Answer :

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]