To find the compound interest earned on a four-year investment of [tex]$3,500 at an annual interest rate of 4.5% compounded monthly, let's go through the detailed steps:
1. Principal, Interest Rate, Time Period, and Compounding Frequency:
- Principal \( P \): $[/tex]3,500
- Annual Interest Rate [tex]\( r \)[/tex]: 4.5% or 0.045 (as a decimal)
- Time [tex]\( t \)[/tex]: 4 years
- Compounding Frequency: Monthly, which is 12 times per year.
2. Monthly Interest Rate:
The monthly interest rate [tex]\( r/n \)[/tex] can be found by dividing the annual interest rate by the number of times it is compounded per year.
[tex]\[
\text{Monthly Interest Rate} = \frac{0.045}{12} = 0.00375
\][/tex]
3. Total Number of Compounding Periods:
The total number of compounding periods [tex]\( n \times t \)[/tex] in four years is:
[tex]\[
\text{Total Number of Periods} = 12 \times 4 = 48
\][/tex]
4. Compound Interest Formula:
Use the compound interest formula to find the final amount [tex]\( A \)[/tex]:
[tex]\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\][/tex]
5. Substituting the Values:
- [tex]\( P = 3500 \)[/tex]
- [tex]\( \frac{r}{n} = 0.00375 \)[/tex]
- [tex]\( nt = 48 \)[/tex]
[tex]\[
A = 3500 \left(1 + 0.00375\right)^{48}
\][/tex]
6. Final Amount:
Using the provided result, the final amount [tex]\( A \)[/tex] is approximately:
[tex]\[
A \approx 4188.85
\][/tex]
7. Interest Earned:
The compound interest earned is the final amount minus the principal:
[tex]\[
\text{Interest Earned} = A - P = 4188.85 - 3500 = 688.85
\][/tex]
So, the compound interest earned on the investment is [tex]\( \$688.85 \)[/tex].
Among the given options, the correct answer is:
[tex]\[
\$688.85
\][/tex]
