To determine the balance after 16 years for an initial deposit of \[tex]$8500 in an account with an annual interest rate of 7% compounded continuously, we can use the formula for continuous compounding. The formula is:
\[
F = Pe^{rt}
\]
where:
- \( P \) is the principal amount (initial deposit),
- \( r \) is the annual interest rate (expressed as a 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),
- \( F \) is the future value of the investment, including interest.
Given the values:
- \( P = 8500 \),
- \( r = 0.07 \),
- \( t = 16 \),
we can plug these values into the formula:
\[
F = 8500 \times e^{0.07 \times 16}
\]
Calculating the exponent first:
\[
0.07 \times 16 = 1.12
\]
Now, we compute \( e^{1.12} \):
\[
e^{1.12} \approx 3.065
\]
Multiplying by the principal:
\[
F \approx 8500 \times 3.065 = 26051.26
\]
Therefore, the balance after 16 years, rounded to the nearest cent, is:
\[
F = \$[/tex]26051.26
\]
So, the balance after 16 years is \$26051.26.
