To determine how much Jenny will have in her account after 4 years, we will use the compound interest formula:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after [tex]\( t \)[/tex] years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money), which is [tex]$4000.
- \( r \) is the annual interest rate (decimal), which is 0.02 (2%).
- \( n \) is the number of times that interest is compounded per year, which is 4 (since it’s compounded quarterly).
- \( t \) is the number of years the money is invested or borrowed for, which is 4 years.
Now, let's apply these values to the formula step by step:
1. Calculate the rate per period:
\[ \frac{r}{n} = \frac{0.02}{4} = 0.005 \]
2. Calculate the total number of compounding periods:
\[ nt = 4 \times 4 = 16 \]
3. Apply these values to the formula:
\[ A = 4000 \left(1 + 0.005\right)^{16} \]
4. Compute the amount inside the parentheses first:
\[ 1 + 0.005 = 1.005 \]
5. Raise this value to the 16th power:
\[ 1.005^{16} \approx 1.08307153865 \]
6. Multiply this result by the principal:
\[ A = 4000 \times 1.08307153865 = 4332.284605104088 \]
So, the total amount in Jenny's account after 4 years is $[/tex]4332.284605104088.
Since we need to round this to the nearest cent, we get:
[tex]\[ A \approx 4332.28 \][/tex]
Jenny will have approximately $4332.28 in her account after 4 years.
