The maintenance costs associated with a machine are \(\$ 2,000\) per year for the first 10 years and \(\$ 1,000\) per year thereafter. The machine has an infinite life. If interest is 10 percent, what is the present worth of the annual disbursements? a) \(\$ 16,145\) c) \(\$ 21,300\) b) \(\$ 19,678\) d) \(\$ 92,136\)

\(PV_{10} = 2000 \cdot \frac{1 - (1.1)^{-10}}{0.10}\) \(PV_{10} = 2000 \cdot \frac{1 - 0.3855}{0.10}\) \(PV_{10} = 2000 \cdot \frac{0.6145}{0.10}\) \(PV_{10} = 2000 \cdot 6.145\) \(PV_{10} = 12,290\) The present worth of the annual disbursements for the first 10 years is $12,290. #tag_title#Step 2: Calculate the present worth for the remaining years#tag_content# Next, we need to calculate the present worth of the $1,000 annual disbursements for the remaining years (from years 11 to 20). We can use the same present value formula. However, since this annuity starts from the 11th year, we need to discount its present value by 10 years. The present value of these disbursements in year 10 can be calculated as: Given, \(P = 1000\), \(r = 0.10\), and \(n = 10\). Plugging in the values, we have: \(PV_{10\_20} = 1000 \cdot \frac{1 - (1 + 0.10)^{-10}}{0.10}\) Calculating the present value in year 10: \(PV_{10\_20} = 1000 \cdot \frac{1 - (1.1)^{-10}}{0.10}\) \(PV_{10\_20} = 1000 \cdot \frac{1 - 0.3855}{0.10}\) \(PV_{10\_20} = 1000 \cdot \frac{0.6145}{0.10}\) \(PV_{10\_20} = 1000 \cdot 6.145\) \(PV_{10\_20} = 6,145\) Now, we need to discount this value by 10 years to find the present worth in year 0, using the formula: \(PV = \frac{FV}{(1 + r)^n}\), where \(FV\) is the future value. Given, \(FV = 6145\), \(r = 0.10\), and \(n = 10\). Plugging in the values, we have: \(PV_{0\_20} = \frac{6145}{(1 + 0.10)^{10}}\) Calculating the present value in year 0: \(PV_{0\_20} = \frac{6145}{(1.1)^{10}}\) \(PV_{0\_20} = \frac{6145}{2.5937}\) \(PV_{0\_20} = 2368.68\) The present worth of the annual disbursements for the remaining years (11 to 20) is $2,368.68. #tag_title#Step 3: Calculate the total present worth of annual disbursements#tag_content# Finally, to find the total present worth of annual disbursements, we need to add the present worths calculated in the previous steps: Total present worth = Present worth (first 10 years) + Present worth (remaining years) Total present worth = $12,290 + $2,368.68 Total present worth = $14,658.68 The total present worth of annual disbursements for the machine is $14,658.68.