To calculate the total amount Walter will pay for his [tex]$\$[/tex] 6,000[tex]$ loan over six years with an annual interest rate of 6%, compounded annually, we will use the compound interest formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Here,
- \( P \) is the principal amount (initial amount of the loan), which is \$[/tex]6,000.
- [tex]\( r \)[/tex] is the annual interest rate, which is 6% or 0.06.
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year, which in this case is 1 (annually).
- [tex]\( t \)[/tex] is the number of years the money is invested or borrowed for, which is 6 years.
Using these values, the formula simplifies to:
[tex]\[ A = 6000 \left(1 + 0.06\right)^6 \][/tex]
First, we calculate the growth factor:
[tex]\[ 1 + 0.06 = 1.06 \][/tex]
Next, we raise this growth factor to the power of 6:
[tex]\[ 1.06^6 = 1.4185 \][/tex]
Now we multiply the principal by this value:
[tex]\[ A = 6000 \times 1.4185 = 8511.12 \][/tex]
Therefore, the correct total amount Walter will pay after six years is:
[tex]\[ \boxed{8511.12} \][/tex]
Thus, the correct answer is:
C. [tex]$\$[/tex] 8511.12$
