**Rewritten Question:** What is the minimum required annual revenue that will make the purchase of the tabletop CNC mill a break-even investment for your team, given a Minimum Attractive Rate of Return (MARR) of 15%? Use the following data: - Investment Cost: $13,000 - Useful Life: 15 years - Market (Salvage) Value: $3,000 (at end of 15 years) - Annual Operating Expenses: $100 - Overhaul Cost at end of Year 5: $200 - Overhaul Cost at end of Year 10: $550 --- ## Step-by-Step Solution ### **Step 1: List Cash Flows** - **Year 0:** Investment Cost = \(-\$13,000\) - **Years 1–15:** Annual Revenue = \(R\) - **Years 1–15:** Annual Operating Expense = \(-\$100\) - **Year 5:** Overhaul Cost = \(-\$200\) - **Year 10:** Overhaul Cost = \(-\$550\) - **Year 15:** Salvage Value = \(+\$3,000\) ### **Step 2: Set Up Net Present Worth (NPW) Equation** For break-even, NPW = 0 at MARR = 15%. \[ NPW = -13,000 + \left[\sum_{n=1}^{15} (R - 100) \cdot (P/A, 15\%, 15)\right] - 200(P/F, 15\%, 5) - 550(P/F, 15\%, 10) + 3,000(P/F, 15\%, 15) \] Where: - \((P/A, 15\%, 15)\) = Present Worth Factor for an annuity (15 years, 15%) - \((P/F, 15\%, n)\) = Present Worth Factor for a single future sum (n years, 15%) ### **Step 3: Find Present Worth Factors** - \((P/A, 15\%, 15)\): \[ (P/A, i, n) = \frac{1 - (1+i)^{-n}}{i} \] \[ (P/A, 15\%, 15) = \frac{1 - (1+0.15)^{-15}}{0.15} \] \[ (1+0.15)^{-15} \approx 0.138 \] \[ 1 - 0.138 = 0.862 \] \[ 0.862 / 0.15 \approx 5.746 \] - \((P/F, 15\%, 5)\): \[ (P/F, 15\%, 5) = (1+0.15)^{-5} \approx 0.497 \] - \((P/F, 15\%, 10)\): \[ (P/F, 15\%, 10) = (1+0.15)^{-10} \approx 0.247 \] - \((P/F, 15\%, 15)\): \[ (P/F, 15\%, 15) = (1+0.15)^{-15} \approx 0.138 \] ### **Step 4: Substitute Values** \[ 0 = -13,000 + (R - 100) \cdot 5.746 - 200 \cdot 0.497 - 550 \cdot 0.247 + 3,000 \cdot 0.138 \] Calculate the constants: - \(200 \cdot 0.497 = 99.4\) - \(550 \cdot 0.247 = 135.85\) - \(3,000 \cdot 0.138 = 414\) \[ 0 = -13,000 + 5.746(R - 100) - 99.4 - 135.85 + 414 \] \[ 0 = -13,000 + 5.746R - 574.6 - 99.4 - 135.85 + 414 \] \[ 0 = -13,000 - 574.6 - 99.4 - 135.85 + 414 + 5.746R \] Calculate the sum of the constants: \[ -13,000 - 574.6 - 99.4 - 135.85 + 414 = -13,000 - 574.6 - 99.4 - 135.85 + 414 \] \[ = -13,000 - 574.6 - 99.4 - 135.85 + 414 \] \[ = -13,000 - 574.6 = -13,574.6 \] \[ -13,574.6 - 99.4 = -13,674 \] \[ -13,674 - 135.85 = -13,809.85 \] \[ -13,809.85 + 414 = -13,395.85 \] So: \[ 0 = -13,395.85 + 5.746R \] \[ 5.746R = 13,395.85 \] \[ R = \frac{13,395.85}{5.746} \approx 2,332 \] --- ### **Final Answer** **The minimum required Annual Revenue to make the purchase profitable at a 15% MARR is approximately:** \[ \boxed{\$2,332 \text{ per year}} \]