1. **State the problem:** Calculate the profitability index (PI) and the modified benefit/cost (B/C) ratio for a wind turbine project with given cash flows, a MARR of 10%, and a project life of 25 years. 2. **Given data:** - Benefits: $20,000 at year 0 and $30,000 at year 5 - Savings: $2,000 per year from years 1 to 20 - Cost: $50,000 at year 0 - Disbenefits: $3,000 per year from years 1 to 10 - MARR (i) = 10% = 0.10 - Project life = 25 years 3. **Formulas:** - Present Value (PV) of a single future amount: $$PV = \frac{F}{(1+i)^n}$$ - Present Value of an annuity (uniform series) from year 1 to N: $$PV = A \times \frac{1 - (1+i)^{-N}}{i}$$ - Profitability Index (PI): $$PI = \frac{PV(benefits)}{PV(costs)}$$ - Modified Benefit/Cost ratio (B/C): $$B/C = \frac{PV(benefits) + PV(savings)}{PV(costs) + PV(disbenefits)}$$ 4. **Calculate PV of benefits:** - Benefit at year 0: $20,000$ (already present value) - Benefit at year 5: $$PV = \frac{30,000}{(1+0.10)^5} = \frac{30,000}{1.61051} \approx 18,626.9$$ - Savings from year 1 to 20: annuity of $2,000$ per year $$PV = 2,000 \times \frac{1 - (1.10)^{-20}}{0.10} = 2,000 \times 8.5136 = 17,027.2$$ - Total PV benefits = $20,000 + 18,626.9 + 17,027.2 = 55,654.1$ 5. **Calculate PV of costs:** - Cost at year 0: $50,000$ (already present value) - Disbenefits from year 1 to 10: annuity of $3,000$ per year $$PV = 3,000 \times \frac{1 - (1.10)^{-10}}{0.10} = 3,000 \times 6.1446 = 18,433.8$$ - Total PV costs = $50,000 + 18,433.8 = 68,433.8$ 6. **Calculate Profitability Index (PI):** $$PI = \frac{55,654.1}{50,000} = 1.1131$$ (Note: PI uses only initial cost, not disbenefits) 7. **Calculate Modified Benefit/Cost ratio (B/C):** $$B/C = \frac{55,654.1}{68,433.8} = 0.813$$ **Final answers:** - Profitability Index (PI) = 1.11 (rounded to two decimals) - Modified Benefit/Cost ratio (B/C) = 0.81 (rounded to two decimals)