Final answer:
To calculate the initial deposit needed to have $10,000 after ten years with a 10% annual compound interest rate, we use the compound interest formula and find that approximately $3,855.43 needs to be deposited.
Step-by-step explanation:
The student is asking about the concept of compound interest, which is a fundamental topic in mathematics and finance. To answer the question of how much money needs to be deposited into a bank account with a 10% interest rate compounded annually to have $10,000 after ten years, we use the compound interest formula:
P = A / (1 + r)^n
Where:
- P is the initial principal balance (the amount to find out)
- A is the future value of the investment/loan, including interest
- r is the annual interest rate (decimal)
- n is the number of years the money is invested/borrowed for
Putting the values from the question into the formula, we calculate:
$10,000 = P * (1 + 0.10)^10
We then solve for P:
P = $10,000 / (1.10)^10
P = $10,000 / 2.59374
P ≈ $3,855.43
Thus, approximately $3,855.43 needs to be deposited into the bank account to have $10,000 in ten years at a 10% interest rate compounded annually.