Let's break down the problem step by step: ### **Given Data** - **Years 1–4:** Cost = $1.8 million per year - **Year 5:** Cost = $1.77 million - **Year 6:** Cost = $1.74 million - **Years 7–10:** Cost decreases by $30,000 ($0.03 million) each year - Year 7: $1.71 million - Year 8: $1.68 million - Year 9: $1.65 million - Year 10: $1.62 million - **Interest Rate:** 15% per year --- ### **Step 1: List All Cash Flows** | Year | Cost ($ million) | |------|------------------| | 1 | 1.80 | | 2 | 1.80 | | 3 | 1.80 | | 4 | 1.80 | | 5 | 1.77 | | 6 | 1.74 | | 7 | 1.71 | | 8 | 1.68 | | 9 | 1.65 | | 10 | 1.62 | --- ### **Step 2: Present Worth (PW) Formula** PW = \( \sum_{n=1}^{10} \frac{C_n}{(1+i)^n} \) Where: - \( C_n \) = cash flow in year n - \( i \) = 15% = 0.15 --- ### **Step 3: Calculate Each Present Worth** Let's do each year: #### Years 1–4: For years 1–4, the cost is $1.8 million each year. \[ PW_{1-4} = 1.8 \left( \frac{1}{1.15^1} + \frac{1}{1.15^2} + \frac{1}{1.15^3} + \frac{1}{1.15^4} \right) \] #### Years 5–10: Apply the formula for each year. \[ PW_5 = \frac{1.77}{1.15^5} \] \[ PW_6 = \frac{1.74}{1.15^6} \] \[ PW_7 = \frac{1.71}{1.15^7} \] \[ PW_8 = \frac{1.68}{1.15^8} \] \[ PW_9 = \frac{1.65}{1.15^9} \] \[ PW_{10} = \frac{1.62}{1.15^{10}} \] --- ### **Step 4: Plug Numbers and Sum Up** Let's calculate each term: #### Years 1–4 \[ \frac{1}{1.15^1} = 0.8696 \] \[ \frac{1}{1.15^2} = 0.7561 \] \[ \frac{1}{1.15^3} = 0.6575 \] \[ \frac{1}{1.15^4} = 0.5718 \] Sum: \( 0.8696 + 0.7561 + 0.6575 + 0.5718 = 2.855 \) \[ PW_{1-4} = 1.8 \times 2.855 = 5.139 \] --- #### Years 5–10 - \( 1.15^5 = 2.0114 \implies 1/2.0114 = 0.497 \) - \( 1.15^6 = 2.3131 \implies 1/2.3131 = 0.432 \) - \( 1.15^7 = 2.6601 \implies 1/2.6601 = 0.376 \) - \( 1.15^8 = 3.0591 \implies 1/3.0591 = 0.327 \) - \( 1.15^9 = 3.517 \implies 1/3.517 = 0.284 \) - \( 1.15^{10} = 4.0446 \implies 1/4.0446 = 0.247 \) So: \[ PW_5 = 1.77 \times 0.497 = 0.879 \] \[ PW_6 = 1.74 \times 0.432 = 0.753 \] \[ PW_7 = 1.71 \times 0.376 = 0.643 \] \[ PW_8 = 1.68 \times 0.327 = 0.549 \] \[ PW_9 = 1.65 \times 0.284 = 0.469 \] \[ PW_{10} = 1.62 \times 0.247 = 0.400 \] Sum: \( 0.879 + 0.753 + 0.643 + 0.549 + 0.469 + 0.400 = 3.693 \) --- ### **Step 5: Total Present Worth** \[ PW_{\text{total}} = PW_{1-4} + PW_{5-10} = 5.139 + 3.693 = \boxed{8.83 \text{ million}} \] --- ## **Final Answer** \[ \boxed{8.83 \text{ million}} \] This is the present worth of the pumping costs at year 0 at an interest rate of 15% per year.