Final answer:
To find the amount owed after 8 years on a $2000 loan with 10% interest compounded quarterly, we use the compound interest formula A = P(1 + r/n)^(nt). Substituting the values, we find A = 2000(1.025)^32, which calculates the future value of the loan.
Step-by-step explanation:
The student's question is about determining the future value of an investment with compound interest. To calculate the amount owed after 8 years when $2000 is loaned at a 10% annual interest rate, compounded quarterly, we use 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 sum of money), r is the annual interest rate (decimal), n is the number of times that interest is compounded per unit t, and t is the time the money is invested for in years.
In this case, we have P = $2000, r = 0.10 (as 10%), n = 4 (compounded quarterly), and t = 8 years. Plugging these into the formula, we get:
A = 2000(1 + 0.10/4)^(4*8)
A = 2000(1 + 0.025)^(32)
A = 2000(1.025)^32
After calculating the above expression, we find the total amount that will be owed after 8 years.
By applying this formula, the process of compounding interest will show that the investment grows at a faster rate than simple interest, because interest is earned not only on the original principal but also on accumulated interest from previous periods. This is an important concept in finance and affects both investments and loans.