Final answer:
The continuously compounded interest over eight years for a $10,000 investment at a 5.5% interest rate is calculated using the formula A = Pe^(rt), where P is the principal amount, r is the annual interest rate, t is the time in years, and e is the base of the natural logarithm.
Step-by-step explanation:
To calculate the continuously compounded interest on a principal balance of $10,000 at an interest rate of 5.5% over eight years, we use the formula for continuously compounded interest: A = Pert, where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate (in decimal form), t is the time the money is invested for in years, and e is the base of the natural logarithm, approximately equal to 2.71828.
First, we convert the interest rate from a percentage to a decimal by dividing by 100: 5.5% ÷ 100 = 0.055.
Then we apply the values to the formula: A = 10,000 × e(0.055 × 8).
After doing the calculation with the values of e, r, and t, we find the value of A, which is the total amount after 8 years. To find just the interest earned, subtract the original principal from this amount: Interest = A - P. Note that this will give a value greater than what you would get with simple interest or compound interest calculated on an annual basis due to the nature of continuous compounding, which compounds interest at every possible moment.