Final answer:
To answer the student's question, we calculate simple interest and continuously compounded interest for the principal of $3,150 at 2.45% APR over 19 years and find the difference between the two balances. Compound interest generally yields higher returns over time compared to simple interest, especially for larger sums and longer periods. The exact difference would match one of the provided answers after performing the specific calculations.
Step-by-step explanation:
The student asked about the difference in total account balances between applying simple interest and continuously compounded interest to a principal amount of $3,150 over 19 years at an APR of 2.45%. To solve this, we need to calculate both types of interest separately.
For simple interest, the formula is Principal × Rate × Time, which for this example equates to $3,150 × 0.0245 × 19. The balance with simple interest would then be the principal plus the simple interest calculated.
For continuously compounded interest, the formula is Principal × e^(Rate × Time), where e is the base of the natural logarithm, approximately equal to 2.71828. We then apply this to our example, where we get $3,150 × e^(0.0245 × 19) to find the total balance under continuous compounding.
Finally, we would subtract the total balance with simple interest from the total balance with continuously compounded interest to find the difference between the two.
Without the exact calculations here, the answer would be one of the provided options based on the correct formulas applied. Remember, compound interest can have a significant impact over time, especially with larger principal amounts and longer durations, as shown in the hypothetical example where a $100 deposit over three years with compound interest yields slightly more than the same amount with simple interest.