Final answer:
To determine the initial investment needed for an account to grow to $10,000 with compound interest, formulas for both annual and continuous compounding are utilized, with the key variables including the principal amount, interest rate, compounding frequency, and investment duration.
Step-by-step explanation:
To calculate how much money one must put into a bank account with a given interest rate to achieve a certain amount in the future, we use the formula for compound interest. Given a 10% interest rate compounded annually, we want to find out the present value that will become $10,000 in ten years.
The formula for compound interest is:
A = P(1 + r/n)^(nt)
Where:
A = the amount of money accumulated after n years, including interest.
P = the principal amount (the initial amount of money)
r = the annual interest rate (decimal)
n = the number of times that interest is compounded per year
t = the number of years the money is invested for
For continuous compounding, the formula is:
A = Pe^(rt)
Where:
e = the base of the natural logarithm
r = the annual interest rate (as a decimal)
t = the time in years
To answer the question, we would need the initial investment amount (P) for both monthly and continuous compounding scenarios to achieve $10,000 after 30 years, and then find the difference between the two results.