When comparing two alternatives with different useful lives, the most appropriate method is to use the Equivalent Uniform Annual Value (EUAV). This method converts the net present worth (NPW) or the total cost of each alternative into an equivalent uniform annual cost or value over their respective lives. By converting the costs into a common annual value, it allows for a straightforward comparison between alternatives with differing lifespans, ensuring that all costs (including maintenance, operation, and initial costs) are considered over a consistent time period.