To determine how much money will be in the Laffite family's account after 25 years given that they deposit \[tex]$8,500 at an annual interest rate of 6.75% compounded continuously, we use the formula for continuous compounding:
\[ A = P \cdot e^{rt} \]
where:
- \( A \) is the amount of money accumulated after \( t \) years, including interest.
- \( P \) is the principal amount (the initial deposit).
- \( r \) 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:
- \( P = 8500 \) (the initial deposit)
- \( r = 6.75\% = 0.0675 \) (the annual interest rate in decimal form)
- \( t = 25 \) years
Substituting these values into the continuous compounding formula:
\[ A = 8500 \cdot e^{(0.0675 \cdot 25)} \]
To find \( e^{(0.0675 \cdot 25)} \):
1. Calculate \( 0.0675 \times 25 \):
\[ 0.0675 \times 25 = 1.6875 \]
2. Now calculate \( e^{1.6875} \):
\[ e^{1.6875} \approx 5.4057 \] (This is a rounded approximation. The exact value can be calculated using a scientific calculator or software.)
3. Substitute \( e^{1.6875} \) back into the formula:
\[ A = 8500 \cdot 5.4057 \]
4. Finally, calculate the amount \( A \):
\[ A \approx 8500 \times 5.4057 = 45,950.57 \]
Therefore, the amount in the account after 25 years will be \$[/tex]45,950.57.
Thus, the closest answer to our calculated value is:
[tex]\[ \boxed{45,950.57} \][/tex]
