Let's solve the problem step by step. ### **Step 1: Find Annual Depreciation for Each System** #### **System A** - Initial Cost = \$290,000 - Life = 4 years - Salvage Value = \$0 > **Annual Depreciation (A) = (Cost - Salvage) / Life = \$290,000 / 4 = \$72,500** #### **System B** - Initial Cost = \$370,000 - Life = 6 years - Salvage Value = \$0 > **Annual Depreciation (B) = (Cost - Salvage) / Life = \$370,000 / 6 = \$61,666.67** --- ### **Step 2: Calculate Annual After-Tax Operating Cost** #### **System A** - Operating Cost = \$93,000/year - Depreciation = \$72,500/year - Tax Rate = 25% > **Tax shield from depreciation** = Depreciation × Tax Rate > = \$72,500 × 0.25 = \$18,125 > **After-tax Operating Cost** = Operating Cost × (1 - Tax Rate) + Depreciation × Tax Rate > = \$93,000 × 0.75 + \$72,500 × 0.25 > = \$69,750 + \$18,125 > = \$87,875 #### **System B** - Operating Cost = \$87,000/year - Depreciation = \$61,666.67/year > **Tax shield from depreciation** = \$61,666.67 × 0.25 = \$15,416.67 > **After-tax Operating Cost** = \$87,000 × 0.75 + \$61,666.67 × 0.25 > = \$65,250 + \$15,416.67 > = \$80,666.67 --- ### **Step 3: Calculate the Present Value (PV) of Costs for Each System** #### **System A** - Initial cost: \$290,000 (at time 0) - After-tax annual cost: \$87,875 (for 4 years) - Discount rate: 8% PV of annual costs (ordinary annuity): \[ PV_{\text{operating}} = C \times \left[ \frac{1 - (1 + r)^{-n}}{r} \right] \] \[ PV_{\text{operating, A}} = 87,875 \times \left[ \frac{1 - (1 + 0.08)^{-4}}{0.08} \right] \] \[ (1 + 0.08)^{-4} = (1.08)^{-4} \approx 0.73503 \] \[ 1 - 0.73503 = 0.26497 \] \[ 0.26497 / 0.08 = 3.3121 \] \[ PV_{\text{operating, A}} = 87,875 \times 3.3121 = \$290,976.84 \] \[ PV_{\text{total, A}} = \text{Initial Cost} + PV_{\text{operating, A}} = 290,000 + 290,976.84 = \$580,976.84 \] --- #### **System B** - Initial cost: \$370,000 - After-tax annual cost: \$80,666.67 (for 6 years) - Discount rate: 8% \[ PV_{\text{operating, B}} = 80,666.67 \times \left[ \frac{1 - (1 + 0.08)^{-6}}{0.08} \right] \] \[ (1 + 0.08)^{-6} = (1.08)^{-6} \approx 0.63017 \] \[ 1 - 0.63017 = 0.36983 \] \[ 0.36983 / 0.08 = 4.6229 \] \[ PV_{\text{operating, B}} = 80,666.67 \times 4.6229 = \$372,939.60 \] \[ PV_{\text{total, B}} = \text{Initial Cost} + PV_{\text{operating, B}} = 370,000 + 372,939.60 = \$742,939.60 \] --- ### **Step 4: Find the Equivalent Annual Cost (EAC) for Each System** EAC is the annuity payment with the same PV over the system's life. \[ EAC = \frac{PV}{\text{Annuity Factor}} \] Annuity factor \( = \frac{1 - (1+r)^{-n}}{r} \) #### **System A: n = 4, r = 0.08** \[ \text{Annuity Factor}_A = 3.3121 \] \[ EAC_A = \frac{580,976.84}{3.3121} = \$175,385.41 \] #### **System B: n = 6, r = 0.08** \[ \text{Annuity Factor}_B = 4.6229 \] \[ EAC_B = \frac{742,939.60}{4.6229} = \$160,732.79 \] --- ## **Final Answers (rounded to 2 decimal places):** \[ \boxed{ \begin{align*} \text{EAC for System A} &= \$175,385.41 \\ \text{EAC for System B} &= \$160,732.79 \\ \end{align*} } \] **System B has the lower equivalent annual cost and is the better choice.**