Certainly! To determine how much money you would have after investing [tex]$1800 at a 4% annual interest rate, compounded monthly, over a period of 6 years, we can use the compound interest formula. The formula for compound interest is given by:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money). In this case, \( P = 1800 \) dollars.
- \( r \) is the annual interest rate (decimal). Here, \( r = 0.04 \).
- \( n \) is the number of times interest is compounded per year. For this problem, \( n = 12 \) (monthly compounding).
- \( t \) is the time the money is invested for, in years. Here, \( t = 6 \).
Let's break down the process:
1. Convert the annual interest rate to a monthly rate:
\[ \frac{r}{n} = \frac{0.04}{12} = 0.0033333 \]
2. Calculate how many times the interest will be compounded over the entire period:
\[ nt = 12 \times 6 = 72 \]
3. Substitute \( P \), \( \frac{r}{n} \), and \( nt \) into the compound interest formula:
\[ A = 1800 \left(1 + 0.0033333\right)^{72} \]
4. Calculate the amount \( A \):
\[ A \approx 2287.34 \]
After 6 years, you would have approximately $[/tex]2287.34.
Additionally, we can determine the interest earned during this time by subtracting the initial principal from the final amount:
[tex]\[ \text{Interest Earned} = A - P = 2287.34 - 1800 = 487.34 \][/tex]
Therefore, the total amount accumulated is approximately [tex]$2287.34, and the interest earned is approximately $[/tex]487.34.
