Let's break down the problem and solve it step by step. ### Given Data #### Hydraulic Press (HP) - First cost: **$680,000** - Annual cost: **$200,000** - Salvage value: **$70,000** - Life: **5 years** #### Electric Machine Technology (EMT) - First cost: **$800,000** - Annual cost: **$152,000** - Salvage value: **$130,000** - Life: **5 years** ### Objective Use an **Annual Worth (AW) equation** to determine the **Rate of Return (ROR)** on the extra investment required for the EMT alternative. --- ## Step 1: Calculate the Difference (Incremental Analysis) ### Incremental Initial Cost (ΔP) \[ \Delta P = 800,000 - 680,000 = \$120,000 \] ### Incremental Annual Cost (ΔA) \[ \Delta A = 152,000 - 200,000 = -\$48,000 \] ### Incremental Salvage Value (ΔS) \[ \Delta S = 130,000 - 70,000 = \$60,000 \] --- ## Step 2: Set up the AW Equation The incremental cash flows: - **Initial cost:** -$120,000 (at year 0) - **Annual savings:** +$48,000 (every year for 5 years) - **Salvage value:** +$60,000 (at year 5) The AW equation (at ROR \( i \)) for 5 years is: \[ 0 = -\Delta P + (\Delta A) \cdot (P/A, i, 5) + \Delta S \cdot (P/F, i, 5) \] \[ 0 = -120,000 + 48,000 \cdot (P/A, i, 5) + 60,000 \cdot (P/F, i, 5) \] --- ## Step 3: Substitute Factors - \( (P/A, i, 5) \) = Present worth factor for annuity - \( (P/F, i, 5) \) = Present worth factor for single sum Formulas: - \( (P/A, i, n) = \frac{1 - (1 + i)^{-n}}{i} \) - \( (P/F, i, n) = (1 + i)^{-n} \) So, \[ 0 = -120,000 + 48,000 \left[\frac{1 - (1 + i)^{-5}}{i}\right] + 60,000 (1 + i)^{-5} \] --- ## Step 4: Solve for \( i \) (Using Trial and Error or Financial Calculator) This is a nonlinear equation, typically solved by trial and error or with a financial calculator/spreadsheet. **Let’s estimate \( i \):** ### Try \( i = 20\% \) (0.20): - \( (P/A, 20\%, 5) = \frac{1 - (1 + 0.20)^{-5}}{0.20} = \frac{1 - 0.4019}{0.20} = 2.990 \) - \( (P/F, 20\%, 5) = (1.20)^{-5} = 0.4019 \) Plug in values: \[ 0 = -120,000 + 48,000 \cdot 2.990 + 60,000 \cdot 0.4019 \] \[ 0 = -120,000 + 143,520 + 24,114 = 47,634 \] Since the result is positive, try a **higher rate**. ### Try \( i = 35\% \) (0.35): - \( (P/A, 35\%, 5) = \frac{1 - (1 + 0.35)^{-5}}{0.35} = \frac{1-0.2401}{0.35} = 2.171 \) - \( (P/F, 35\%, 5) = (1.35)^{-5} = 0.2401 \) Plug in: \[ 0 = -120,000 + 48,000 \cdot 2.171 + 60,000 \cdot 0.2401 \] \[ 0 = -120,000 + 104,208 + 14,406 = -1,386 \] Now we're very close to zero, but slightly negative, so the answer is **slightly less than 35%**. ### Interpolate between 30% and 35%: #### Try \( i = 33\% \) (0.33): - \( (1.33)^{-5} = 0.2617 \) - \( (P/A, 33\%, 5) = \frac{1 - 0.2617}{0.33} = 2.235 \) Plug in: \[ 0 = -120,000 + 48,000 \cdot 2.235 + 60,000 \cdot 0.2617 \] \[ 0 = -120,000 + 107,280 + 15,702 = 2,982 \] So, the ROR is somewhere between 33% and 35%. ### Linear interpolation: \[ i = 33\% + \frac{2,982}{2,982 + 1,386} \times (35\%-33\%) \] \[ = 33\% + \frac{2,982}{4,368} \times 2\% \] \[ = 33\% + 0.6825 \times 2\% \approx 33\% + 1.37\% = 34.4\% \] --- ## **Final Answer** **The ROR on the extra investment for the EMT alternative is approximately \( \boxed{34\%} \).**