1. **Problem Statement:** Kristen invested 12000 in a fund earning 5.5% compounded monthly. He withdraws 800 at the end of every quarter starting 5 years from now. We need to find how long it will take for the fund to be depleted. 2. **Relevant Formula:** This is a problem involving the depletion of an investment with periodic withdrawals and compound interest. The formula for the present value of an annuity with withdrawals is: $$P = W \times \frac{1 - (1 + i)^{-n}}{i}$$ where: - $P$ is the initial principal (12000), - $W$ is the withdrawal amount per period (800), - $i$ is the interest rate per period, - $n$ is the number of withdrawal periods. 3. **Calculate the interest rate per quarter:** Annual nominal rate = 5.5% compounded monthly Monthly rate $i_m = \frac{5.5}{100 \times 12} = 0.0045833$ Quarterly rate $i = (1 + i_m)^3 - 1 = (1 + 0.0045833)^3 - 1$ Calculate: $$i = 1.0045833^3 - 1 = 1.0138 - 1 = 0.0138$$ 4. **Set up the equation:** The withdrawals start after 5 years, so the fund grows for 5 years without withdrawals. Number of months in 5 years = 60 Value after 5 years: $$P_5 = 12000 \times (1 + i_m)^{60} = 12000 \times 1.0045833^{60}$$ Calculate: $$P_5 = 12000 \times 1.3086 = 15703.2$$ 5. **Use the annuity formula to find $n$:** $$15703.2 = 800 \times \frac{1 - (1 + 0.0138)^{-n}}{0.0138}$$ Divide both sides by 800: $$\frac{15703.2}{800} = \frac{1 - (1 + 0.0138)^{-n}}{0.0138}$$ $$19.629 = \frac{1 - (1.0138)^{-n}}{0.0138}$$ Multiply both sides by 0.0138: $$19.629 \times 0.0138 = 1 - (1.0138)^{-n}$$ $$0.2709 = 1 - (1.0138)^{-n}$$ Rearranged: $$(1.0138)^{-n} = 1 - 0.2709 = 0.7291$$ 6. **Solve for $n$:** Take natural logarithm: $$-n \ln(1.0138) = \ln(0.7291)$$ $$-n \times 0.0137 = -0.316$$ $$n = \frac{0.316}{0.0137} = 23.06$$ 7. **Interpretation:** $n$ is the number of quarters withdrawals last. Convert quarters to years and months: $$\text{Years} = \lfloor \frac{23.06}{4} \rfloor = 5$$ $$\text{Months} = (23.06 - 5 \times 4) \times 3 = 3.18 \approx 4 \text{ months (rounded up)}$$ 8. **Total time until depletion:** Initial 5 years growth + 5 years 4 months withdrawals = 10 years 4 months. **Final answer:** The fund will be depleted after **10 years and 4 months** from the initial investment.