Final answer:
The future value for an investment of $8906.54 at 3% interest compounded semiannually for 9 years is approximately $11962.17, and the interest earned is $3055.63. For continuous compounding, the future value is approximately $12017.48, with interest earned being $3110.94.
Step-by-step explanation:
Future Value and Interest Earned for Compounded Investments
To calculate the future value of an investment of $8906.54 for 9 years at 3% interest compounded semiannually, we use the formula:
A = P(1 + r/n)nt
Where:
- A is the future value of the investment/loan, including interest
- P is the principal investment amount ($8906.54)
- 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 or borrowed for, in years
So for semiannual compounding:
- P = $8906.54
- r = 0.03
- n = 2
- t = 9
Plugging in the values, we get:
A = $8906.54(1 + 0.03/2)2*9
Calculating this comes to approximately $11962.17.
The interest earned is the future value minus the initial investment which is $11962.17 - $8906.54 = $3055.63.
For continuous compounding, the formula is:
A = Pert
Where e is the base of the natural logarithm, approximately equal to 2.71828.
So we calculate:
A = $8906.54 * e0.03*9
Which comes to approximately $12017.48, and the interest earned is $12017.48 - $8906.54 = $3110.94.