To determine how much money will be in the account after 25 years with an annual interest rate of 5.25%, we use the compound interest formula, which is given by:
[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).
- [tex]\( r \)[/tex] is the annual interest rate (in decimal form).
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year.
- [tex]\( t \)[/tex] is the number of years the money is invested for.
For this problem, we have:
- [tex]\( P = 1600 \)[/tex] dollars
- [tex]\( r = 5.25\% = \frac{5.25}{100} = 0.0525 \)[/tex]
- [tex]\( t = 25 \)[/tex] years
Assuming the interest is compounded annually ([tex]\( n = 1 \)[/tex]), the formula simplifies to:
[tex]\[ A = P \left(1 + r\right)^{t} \][/tex]
Plugging in the values, we get:
[tex]\[ A = 1600 \left(1 + 0.0525\right)^{25} \][/tex]
[tex]\[ A = 1600 \left(1.0525\right)^{25} \][/tex]
Using this formula, calculate the accumulated amount [tex]\( A \)[/tex] after evaluating the exponent and multiplication steps:
[tex]\[ A \approx 1600 \left(3.593787\right) \][/tex]
[tex]\[ A \approx 5750.06 \][/tex]
Therefore, the amount in the account after 25 years will be approximately \$5750.06, to the nearest cent.
