Final answer:
To compare the two alternatives, we can use the Net Present Worth (NPW) method. Alternative A has a lower NPW compared to alternative B, making it the better choice. Alternative A has an initial cost of $10,000 and an annual maintenance cost of $1,500 for 5 years, while Alternative B has an initial cost of $20,000 and an annual maintenance cost of $1,200 for 5 years.
Step-by-step explanation:
To compare the two alternatives, we can use the Net Present Worth (NPW) method. NPW is the sum of the present values of all the cash flows over the project's life.
For alternative A, the initial cost is $10,000 and there is an annual maintenance cost of $1,500 for 5 years. To calculate the NPW, we need to discount the future cash flows at a rate of 10%. The NPW for alternative A is:
Year 1: -$10,000
Year 2: -$1,500/(1+0.10)^2 = -$1,420.17
Year 3: -$1,500/(1+0.10)^3 = -$1,289.71
Year 4: -$1,500/(1+0.10)^4 = -$1,172.46
Year 5: -$1,500/(1+0.10)^5 = -$1,065.87
Adding up all the present values, we get a total NPW of -$15,947.21.
For alternative B, the initial cost is $20,000 and there is an annual maintenance cost of $1,200 for 5 years. Using the same discount rate of 10%, we can calculate the NPW as:
Year 1: -$20,000
Year 2: -$1,200/(1+0.10)^2 = -$1,081.82
Year 3: -$1,200/(1+0.10)^3 = -$982.56
Year 4: -$1,200/(1+0.10)^4 = -$893.24
Year 5: -$1,200/(1+0.10)^5 = -$812.94
The total NPW for alternative B is -$24,770.56.
Based on the NPW calculations, alternative A has a lower NPW of -$15,947.21 compared to alternative B's NPW of -$24,770.56. Therefore, alternative A is the better choice as it has a lower net present worth and would result in lower costs over the 5-year period.
The question requires comparing two alternatives for solving a valve failure problem using the Net Present Worth method at a 10% discount rate. NPW sums the initial cost and the present worth of future annual maintenance costs over 5 years. The alternative with the lower NPW represents the more cost-efficient choice.
To determine which method is better for solving a valve failure problem using the Net Present Worth (NPW) method, we need to calculate the present worth of each alternative at a discount rate of 10%. The NPW considers both the initial cost and the annual maintenance costs over a 5-year period.
For option A, the initial cost is $10,000 with an annual maintenance cost of $1,500. For option B, the initial cost is $20,000 with an annual maintenance cost of $1,200.
To calculate the NPW, we must discount the future maintenance costs to their present values and add them to the initial cost. The formula to calculate the present value of an annual expense E for n years at a discount rate r is:
NPW = Initial Cost + E × Σfrom year 1 to n (1 / (1 + r)year)
Applying this formula, we can find the NPW for both options and compare them directly. The option with the lower NPW is more cost-efficient.