To determine the balance of a deposit in an account with continuous compounding interest, we use the continuous compounding formula, which is:
[tex]\[ A = P \cdot e^{rt} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after [tex]\( n \)[/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]\( t \)[/tex] is the time the money is invested or borrowed for, in years.
- [tex]\( e \)[/tex] is the base of the natural logarithm, approximately equal to 2.71828.
Given:
- The principal amount ([tex]\( P \)[/tex]) is [tex]$400.
- The annual interest rate (\( r \)) is 5.5%, which in decimal form is \( 0.055 \).
- The time (\( t \)) is 8 years.
Next, we can calculate the balance (\( A \)):
\[ A = 400 \cdot e^{(0.055 \cdot 8)} \]
From the provided continuous compounding formula and given values, we find that when these values are input into the formula, we get:
\[ A \approx \$[/tex]621.08 \]
Therefore, the balance after 8 years will be approximately:
[tex]\[ \boxed{\$621.08} \][/tex]
