Final answer:
The annual compound interest rate (x%) for an investment of $8,000 that grows to $8,877.62 over 6 years is 1.75%, demonstrating the power of compound interest over time.
Step-by-step explanation:
We are tasked with determining the annual compound interest rate (x%) for an investment of $8,000 that grows to $8,877.62 over a period of 6 years. The general formula used for compound interest 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 (the initial amount of money).
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 for, in years.
In our specific scenario, the formula modifies to 8877.62 = 8000(1 + x)^6. We assume that the interest is compounded once per year (n=1), so the new principal is compounded annually.
To solve for x, we first divide both sides by 8000, which gives us (1 + x)^6 = 1.1097025. We then take the sixth root of both sides to isolate (1 + x), which ultimately allows us to solve for x.
After calculating x, we find that the interest rate to two decimal places is approximately 1.75%. This showcases the impressive power of compound interest even at modest rates over several years.