Certainly! Let's go through the compound interest formula step-by-step to determine the balance after 3 years for a deposit of [tex]$500 at an annual interest rate of 4%, compounded annually.
The formula for compound interest is:
\[ V(t) = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( P \) is the initial (principal) investment.
- \( r \) is the annual interest rate (as a decimal).
- \( n \) is the number of times the interest is compounded per year.
- \( t \) is the number of years.
- \( V(t) \) is the value of the investment after \( t \) years.
Given the parameters:
- \( P = 500 \) (the initial deposit),
- \( r = 0.04 \) (4% annual interest rate),
- \( n = 1 \) (compounded annually),
- \( t = 3 \) (3 years),
We can plug these values into the formula:
\[ V(3) = 500 \left(1 + \frac{0.04}{1}\right)^{1 \cdot 3} \]
First, simplify inside the parenthesis:
\[ 1 + \frac{0.04}{1} = 1.04 \]
Now raise \( 1.04 \) to the power of \( 3 \):
\[ 1.04^3 \]
Calculate \( 1.04^3 \):
\[ 1.04^3 = 1.124864 \]
Now multiply this result by the initial principal \( P \):
\[ 500 \times 1.124864 \]
This gives:
\[ 562.432 \]
Rounding to two decimal places (as is standard for currency), we get:
\[ 562.43 \]
Therefore, the balance after 3 years, when the interest is compounded annually, is:
\[ \boxed{562.43} \]
So, based on the given options, the correct answer is:
\[ \$[/tex] 562.43 \]
