Final answer:
The total account balance after 23 years with an initial principal amount of $830 at an annual interest rate of 3.72% compounded continuously is approximately $1,952.23. This is calculated using the formula for continuous compounding, A = Pe^(rt), resulting in $1,952.81 as the closest answer from the provided options.
Step-by-step explanation:
Calculating the Future Value with Continuous Compounding
When calculating the future value of an investment with continuous compounding, we use the formula A = Pert, 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, and e is the base of the natural logarithm, approximately equal to 2.71828.
To find the total account balance after 23 years for an initial principal of $830 at an annual interest rate of 3.72% compounded continuously, we plug these values into the formula:
A = 830 x e(0.0372 x 23)
Which gives us the total future amount. So,
A ≈ 830 x e(0.8556)
By calculating this, we find that the future value A, or the total account balance after 23 years, is approximately:
A ≈ 830 x 2.3522 ≈ $1,952.23
The option that most closely matches this calculation is $1,952.81, which seems to be the correct answer to the question, considering a possible minimal discrepancy due to rounding during the calculation process.