Let's evaluate the compound interest and the amount accumulated using the given values:
- Principal: [tex]\( P = \$2021 \)[/tex]
- Annual interest rate: [tex]\( r = 4.2\% = 0.042 \)[/tex]
- Number of times the interest is compounded per year: [tex]\( n = 12 \)[/tex]
- Time in years: [tex]\( t = 4 \)[/tex]
The formula for finding the accumulated amount [tex]\( A \)[/tex] is:
[tex]\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\][/tex]
First, substitute the values into the formula:
[tex]\[
A = 2021 \left(1 + \frac{0.042}{12}\right)^{12 \times 4}
\][/tex]
Now simplify inside the parentheses:
[tex]\[
\frac{0.042}{12} = 0.0035
\][/tex]
[tex]\[
1 + 0.0035 = 1.0035
\][/tex]
Next, raise this to the power of [tex]\( 12 \times 4 = 48 \)[/tex]:
[tex]\[
A = 2021 \times (1.0035)^{48}
\][/tex]
Calculating the value within the parentheses raised to the 48th power and then multiplying by the principal amount, the accumulated amount [tex]\( A \)[/tex] is approximately:
[tex]\[
A \approx 2390.0137585104058
\][/tex]
Rounding this amount to the nearest cent, we get:
[tex]\[
A \approx 2390.01
\][/tex]
Next, we calculate the interest earned [tex]\( I \)[/tex] using the formula:
[tex]\[
I = A - P
\][/tex]
Substitute the values:
[tex]\[
I = 2390.0137585104058 - 2021 \approx 369.01375851040575
\][/tex]
Rounding this interest to the nearest cent, we obtain:
[tex]\[
I \approx 369.01
\][/tex]
Thus, the accumulated amount after 4 years is \[tex]$2390.01, and the interest earned is \$[/tex]369.01.
