To find the balance after 8 years for an amount of [tex]$1250 deposited in an account with an annual interest rate of 6.5%, compounded continuously, we need to use the formula for continuously compounded interest. The formula is:
\[ A = P \cdot e^{rt} \]
where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (decimal).
- \( t \) is the time the money is invested for in years.
- \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
Given:
- \( P = \$[/tex]1250 \)
- [tex]\( r = \frac{6.5}{100} = 0.065 \)[/tex]
- [tex]\( t = 8 \)[/tex] years
First, we calculate the exponent:
[tex]\[ rt = 0.065 \times 8 = 0.52 \][/tex]
Next, we find [tex]\( e^{0.52} \)[/tex]. Using this value, we can determine the accumulated amount:
[tex]\[ A = 1250 \times e^{0.52} \][/tex]
[tex]\[ A = 1250 \times 1.682 \][/tex] (approximating [tex]\( e^{0.52} \)[/tex] as 1.682)
[tex]\[ A = 2102.534562123608 \][/tex]
Since we want to round to the nearest cent, the final amount [tex]\( A \)[/tex] is:
[tex]\[ F = \$2102.53 \][/tex]
Thus, the balance after 8 years is \$2102.53.
