Final answer:
To find out how much needs to be deposited in an account with 10% annual compound interest to have $10,000 in ten years, the compound interest formula A = P(1 + r/n)nt is used. With compound interest calculated once per year, the principal amount P required is approximately $3,855.43.
Step-by-step explanation:
The question asks how much money needs to be initially deposited in a bank account with an annual compound interest rate of 10% to reach $10,000 in ten years.
To solve this, we can use the formula for compound interest, which is A = P(1 + r/n)nt, where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate (decimal), n is the number of times that interest is compounded per year, and t is the time the money is invested for in years.
In this case, since the interest is compounded annually, n will be 1. Plugging the values into the formula, we get 10,000 = P(1 + 0.10)10.
Simplifying, P = 10,000 / (1 + 0.10)10. Calculating the right-hand side gives us the principal amount needed today to achieve the future value of $10,000 after ten years.
Calculating the exact value, we find P = $3,855.43 approximately.