To calculate the amount in Emily's account after 10 years, we can use the formula for compound interest:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) = the amount of money accumulated after n years, including interest.
- \( P \) = the principal amount (the initial amount of money).
- \( r \) = the annual interest rate (decimal).
- \( n \) = the number of times that interest is compounded per year.
- \( t \) = the number of years the money is invested for.
Given:
- \( P = 5619 \)
- \( r = 4.04% = 0.0404 \) (as a decimal)
- \( n = 4 \) (since the interest is compounded quarterly)
- \( t = 10 \)
Now plugging in these values into the formula:
\[ A = 5619 \left(1 + \frac{0.0404}{4}\right)^{4 \times 10} \]
First, calculate \( \frac{0.0404}{4} \):
\[ \frac{0.0404}{4} = 0.0101 \]
Now, substitute this value into the formula:
\[ A = 5619 \left(1 + 0.0101\right)^{40} \] \[ A = 5619 \left(1.0101\right)^{40} \]
Next, calculate \( (1.0101)^{40} \):
\[ (1.0101)^{40} \approx 1.448886 \]
Now plug this back into the equation for \( A \):
\[ A = 5619 \times 1.448886 \approx 8151.64 \]
Thus, the total amount in the account after 10 years is approximately:
\[ \boxed{8151.64} \]
rounded to the nearest cent.
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