Final answer:
To calculate the future value of an investment with compound interest, use the formula A = P(1 + r/n)^(n*t). For annual compounding, Rachel needs to put $3855.56 into the bank account. For monthly compounding, she needs to put $3825.92. For continuous compounding, she needs to put $3678.79.
Step-by-step explanation:
To calculate the future value of an investment with compound interest, you can use the formula:
A = P(1 + r/n)^(n*t)
Where:
- A is the future value
- P is the principal amount (the initial investment)
- r is the annual interest rate (written as a decimal)
- n is the number of times that interest is compounded per year
- t is the number of years
For Part A, since the interest is compounded annually, n would be 1. Plugging in the values, we have:
A = P(1 + 0.10/1)^(1*10) = P(1 + 0.10)^10 = 10000
Simplifying the equation:
1.10^10 = 10000/P
Using algebra to solve for P:
P = 10000 / 1.10^10 = $3855.56
Therefore, Rachel needs to put $3855.56 into the bank account to have $10,000 in ten years if the interest is compounded annually.
For Part B, since the interest is compounded monthly, n would be 12 (12 months in a year). Plugging in the values, we have:
A = P(1 + 0.10/12)^(12*10) = P(1 + 0.00833)^120 = 10000
1.00833^120 = 10000/P
Using algebra to solve for P:
P = 10000 / 1.00833^120 = $3825.92
Therefore, Rachel needs to put $3825.92 into the bank account to have $10,000 in ten years if the interest is compounded monthly.
For Part C, continuous compounding uses the formula:
A = Pe^(r*t)
Where:
- A is the future value
- P is the principal amount (the initial investment)
- e is Euler's number (approximately 2.71828)
- r is the annual interest rate (written as a decimal)
- t is the number of years
Plugging in the values, we have:
A = P * e^(0.10*10) = 10000
e^(1) = 10000/P
Using algebra to solve for P:
P = 10000 / e^(1) = $3678.79
Therefore, Rachel needs to put $3678.79 into the bank account to have $10,000 in ten years if the interest is compounded continuously.