To determine the interest earned by Kaiden after 8 years with interest compounded weekly, we use the compound interest formula, which is given by:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after [tex]\( t \)[/tex] years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial deposit).
- [tex]\( r \)[/tex] is the annual interest rate (as a decimal).
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year.
- [tex]\( t \)[/tex] is the time the money is invested for in years.
Let's break each element down with the given values:
- Principal, [tex]\( P \)[/tex]: \[tex]$6200
- Annual interest rate, \( r \): 2.77% or 0.0277 as a decimal
- Number of times compounded per year, \( n \): 52 (since interest is compounded weekly)
- Time, \( t \): 8 years
Plugging these values into the compound interest formula, the next step is to calculate the accumulated amount \( A \):
\[ A = 6200 \left(1 + \frac{0.0277}{52}\right)^{52 \times 8} \]
Calculating inside the parentheses first:
\[ \frac{0.0277}{52} \approx 0.0005327 \]
Now, add 1 to this value:
\[ 1 + 0.0005327 \approx 1.0005327 \]
Next, raise this to the power of \( 52 \times 8 = 416 \):
\[ A = 6200 \times (1.0005327)^{416} \]
Rather than calculating this complex power manually, we acknowledge the result provided:
\[ A \approx 7737.59 \]
This amount includes the initial principal plus the earned interest, which means the interest earned is:
\[ \text{Interest earned} = A - P \]
\[ \text{Interest earned} = 7737.59 - 6200 \approx 1537.59 \]
Thus, the interest Kaiden will earn after 8 years is approximately \$[/tex]1537.59. Out of the given choices, the correct answer is:
[tex]\[ \$1537.59 \][/tex]
