Suppose that the number of hours X for which a machine will operate before it fails has a continuous distribution with p.d.f. f (x). Suppose that at the time at which the machine begins operating you must decide when you will return to inspect it. If you return before the machine has failed, you incur a cost of b dollars for having wasted an inspection. If you return after the machine has failed, you incur a cost of c dollars per hour for the length of time during which the machine was not operating after its failure. What is the optimal number of hours to wait before you return for inspection in order to minimize your expected cost?