Final answer:
The question involves calculating the total savings accumulated over 20 years with a continuous compounding interest rate of 5%. While the provided examples suggest using an annual compound interest formula, the original question requires the use of the continuous compounding formula A = Pert to integrate the amounts saved yearly.
Step-by-step explanation:
The student's question deals with increasing savings through compound interest over a period of time. To compute the final savings after 20 years at a 5% interest rate compounded continuously, we use the formula for continuous compounding, which is A = Pert, where P is the principal amount (initial investment), r is the annual interest rate, t is the time in years, and e is the base of the natural logarithm, approximately equal to 2.71828.
However, the formula provided in the information (3,000(1+.07)40 = $44,923), is for compound interest that is not continuous but compounded once per year. If an investor saves a certain dollar amount each year and this is compounded continuously at 5% interest, then each year's savings would grow according to this new continuous compounding formula.
Therefore, if the investor saves $X every year for 20 years at 5% interest compounded continuously, you would need to integrate the continuous compounding formula from 0 to 20 to find the total amount of money accumulated over the 20 years. This integral can be numerically calculated or, for simplicity, the investor could use the given formula yearly and sum the compounded amounts for a close approximation.