$20,000 is invested at a rate of 5% compounded quarterly. Identify the compound interest function to model the situation. Then find the balance after 5 years.

A. [tex]A = 20000(1.0125)^{4t};[/tex] $25,640.74

B. [tex]A = 20000(1.025)^{2t};[/tex] $20,106.73

C. [tex]A = 20000(1.025)^{2};[/tex] $25,601.69

D. [tex]A = 20000(1.0125)^{4};[/tex] $20,460.81

To solve this problem, we need to identify the correct compound interest function and calculate the future balance using the given parameters. Here's a step-by-step explanation:

Understanding Compound Interest:
Compound interest is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods.
Formula for Compound Interest:
The formula to calculate compound interest is given by:
[tex]A = P \left(1 + \frac{r}{n}\right)^{nt}[/tex]
Where:
- [tex]A[/tex] is the future value of the investment/loan, including interest.
- [tex]P[/tex] is the principal investment amount ($20,000).
- [tex]r[/tex] is the annual interest rate (decimal form, so 5% = 0.05).
- [tex]n[/tex] is the number of times that interest is compounded per year (quarterly compounding means [tex]n = 4[/tex]).
- [tex]t[/tex] is the time in years the money is invested for (5 years).
Plug in the Values:
Using the values in the formula, we have:
[tex]A = 20000 \left(1 + \frac{0.05}{4}\right)^{4 \times 5}[/tex]
Simplify the formula:
[tex]A = 20000 \left(1 + 0.0125\right)^{20}[/tex]
[tex]A = 20000 (1.0125)^{20}[/tex]
Calculate the Balance after 5 Years:
Using a calculator or computational tool, compute:
[tex]A = 20000 \times 1.28203723171 \approx 25640.74[/tex]
Therefore, the future balance after 5 years is approximately $25,640.74.
Conclusion:
Based on the choices provided, the correct compound interest function is:
[tex]A = 20000(1.0125)^{4t}[/tex]
And the balance after 5 years is $25,640.74. Therefore, the correct multiple-choice option is: [tex]A = 20000(1.0125)^{4t}; \$25,640.74[/tex].