Final answer:
The question pertains to calculating the deposit amount needed, in addition to an initial $10,000 at a 6% compounded monthly interest rate, to reach a goal of $30,000 in 10 years. It involves the compound interest formula to find the future value of the initial deposit and to solve for the additional deposit, x.
Step-by-step explanation:
The subject of the question is determining the additional amount x that Shawne needs to deposit into an account already holding $10,000, with a 6% interest rate compounded monthly, in order for the account to grow to $30,000 in 10 years. This is a compound interest problem, which uses the compound interest formula A = P(1 + r/n)^(nt), 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).
- n is the number of times that interest is compounded per year.
- t is the time the money is invested for, in years.
First, we find the future value of the initial $10,000 using the formula. Since 5 years have passed, we calculate the amount for the remaining 10 years.
FV1 = 10000(1 + 0.06/12)^(12*10)
We then subtract FV1 from the future total goal of $30,000, which tells us how much more needs to be invested now (x) to reach the total in 10 years. That calculation looks like this:
x = 30000 - FV1
We finally calculate the value of x using the formula for compound interest, taking x as the new principal P for the next 10 years and solving for it.