To determine the amount of money in the account at the end of 6 years when $2000 is deposited at an interest rate of 3%, compounded quarterly, we use the compound interest formula:
[tex]\[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \][/tex]
Where:
- \(P\) is the principal amount (the initial amount of money), which is $2000.
- \(r\) is the annual interest rate (as a decimal), which is 0.03 (since 3% = 0.03).
- \(n\) is the number of times the interest is compounded per year, which is 4 (since it is compounded quarterly).
- \(t\) is the time the money is invested for, in years, which is 6 years.
Now let's plug these values into the formula step-by-step.
1. Determine the value of \( \frac{r}{n} \):
[tex]\[ \frac{r}{n} = \frac{0.03}{4} = 0.0075 \][/tex]
2. Calculate \( n \times t \):
[tex]\[ n \times t = 4 \times 6 = 24 \][/tex]
3. Substitute these values into the compound interest formula:
[tex]\[ A = 2000 \left( 1 + 0.0075 \right)^{24} \][/tex]
4. Simplify inside the parentheses:
[tex]\[ A = 2000 \left( 1.0075 \right)^{24} \][/tex]
5. Compute \( \left( 1.0075 \right)^{24} \):
[tex]\[ 1.0075^{24} \approx 1.1964159 \][/tex]
6. Finally, multiply this value by the principal amount:
[tex]\[ A = 2000 \times 1.1964159 \][/tex]
[tex]\[ A \approx 2392.83 \][/tex]
So, the amount in the account after 6 years will be approximately \$2392.83.
[tex]\[
\boxed{2392.83}
\][/tex]
