Let's break down the question and solve it step by step: ### **Given Data** #### **Hydraulic Press (HP):** - First cost, \( P_{HP} = \$680,000 \) - Annual cost, \( A_{HP} = \$200,000 \) - Salvage value after 5 years, \( S_{HP} = \$70,000 \) #### **Electric Machine Technology (EMT):** - First cost, \( P_{EMT} = \$800,000 \) - Annual cost, \( A_{EMT} = \$152,000 \) - Salvage value after 5 years, \( S_{EMT} = \$130,000 \) ### **Step 1: Calculate the Incremental Cash Flows (EMT - HP)** - **ΔP (Extra Investment):** \( \$800,000 - \$680,000 = \$120,000 \) - **ΔA (Annual Savings):** \( \$200,000 - \$152,000 = \$48,000 \) (EMT saves \$48,000/year) - **ΔS (Extra Salvage):** \( \$130,000 - \$70,000 = \$60,000 \) ### **Step 2: Write the Annual Worth (AW) Equation for the Increment** Set AW = 0 to solve for the ROR (\( i \)): \[ 0 = -\Delta P \cdot (A/P, i, 5) + \Delta A + \Delta S \cdot (A/F, i, 5) \] Plug in the values: \[ 0 = -120{,}000 \cdot (A/P, i, 5) + 48{,}000 + 60{,}000 \cdot (A/F, i, 5) \] Recall: - \( (A/P, i, n) = \frac{i(1+i)^n}{(1+i)^n - 1} \) - \( (A/F, i, n) = \frac{i}{(1+i)^n - 1} \) So, \[ 0 = -120{,}000 \cdot \frac{i(1+i)^5}{(1+i)^5 - 1} + 48{,}000 + 60{,}000 \cdot \frac{i}{(1+i)^5 - 1} \] \[ 0 = \frac{-120{,}000 i (1+i)^5 + 48{,}000 \left[ (1+i)^5 - 1 \right] + 60{,}000 i}{(1+i)^5 - 1} \] But let's use the original AW equation for incremental investment: \[ 0 = -120{,}000 (A/P, i, 5) + 48{,}000 + 60{,}000 (A/F, i, 5) \] ### **Step 3: Use Trial and Error or Financial Calculator** #### **Let's estimate the ROR by using an iterative approach:** #### **Guess 1: Try 20%** \[ (A/P, 20\%, 5) = \frac{0.20(1.20^5)}{1.20^5 - 1} = \frac{0.20 \cdot 2.48832}{2.48832 - 1} = \frac{0.4977}{1.48832} = 0.3345 \] \[ (A/F, 20\%, 5) = \frac{0.20}{2.48832 - 1} = \frac{0.20}{1.48832} = 0.1344 \] Plug into the AW equation: \[ 0 = -120,000 \times 0.3345 + 48,000 + 60,000 \times 0.1344 \] \[ 0 = -40,140 + 48,000 + 8,064 = 15,924 \] AW is **positive** at 20%. Try a higher i. --- #### **Guess 2: Try 40%** \[ (1.4^5) = 5.37824 \] \[ (A/P, 40\%, 5) = \frac{0.4 \times 5.37824}{5.37824 - 1} = \frac{2.1513}{4.37824} = 0.4913 \] \[ (A/F, 40\%, 5) = \frac{0.4}{4.37824} = 0.0914 \] \[ 0 = -120,000 \times 0.4913 + 48,000 + 60,000 \times 0.0914 \] \[ 0 = -58,956 + 48,000 + 5,484 = -5,472 \] AW is **negative** at 40%. The ROR is between 20% and 40%. --- #### **Interpolate:** \[ i^* = 20 + \frac{15,924}{15,924 + 5,472} \times (40-20) \] \[ = 20 + \frac{15,924}{21,396} \times 20 \] \[ = 20 + 0.744 \times 20 = 20 + 14.88 = 34.88\% \] --- #### **Check at 35%** \[ (1.35^5) = 4.4817 \] \[ (A/P, 35\%, 5) = \frac{0.35 \times 4.4817}{4.4817 - 1} = \frac{1.5686}{3.4817} = 0.4506 \] \[ (A/F, 35\%, 5) = \frac{0.35}{3.4817} = 0.1006 \] \[ 0 = -120,000 \times 0.4506 + 48,000 + 60,000 \times 0.1006 \] \[ 0 = -54,072 + 48,000 + 6,036 = -36 \] Very close to zero! So **35% is the answer.** --- ## **Final Answer** **The ROR on the extra investment for the EMT alternative is:** \[ \boxed{35\%} \] --- **If you need to show the AW-based equation, it is:** \[ 0 = -\$120,000 (A/P, i, 5) + \$48,000 + \$60,000 (A/F, i, 5) \] where \( i = 35\% \).