1. The problem asks to find the initial investment amount given a final amount of 13288.09 after 11 years with an interest rate of 6%. 2. We assume the interest is compounded annually and use the compound interest formula: $$A = P(1 + r)^t$$ where $A$ is the final amount, $P$ is the principal (initial investment), $r$ is the annual interest rate as a decimal, and $t$ is the time in years. 3. Substitute the known values: $$13288.09 = P(1 + 0.06)^{11}$$ 4. Simplify the expression inside the parentheses: $$13288.09 = P(1.06)^{11}$$ 5. Calculate $(1.06)^{11}$: $$1.06^{11} \approx 1.8983$$ 6. Solve for $P$: $$P = \frac{13288.09}{1.8983} \approx 7000$$ 7. Therefore, Archie initially invested approximately 7000 to the nearest 1.