Renewing highway pavement markings5 costs c dollars per mile but reduces social costs from delays, accidents, and other effects of declining marking performance over time. Suppose that new markings yield maximum performance p max and that performance t days after renewal can be expressed p1t2!p max e-at, where a is a constant depending on the durability of the markings.

Also assume that each unit of lost performance costs d dollars per day per mile. Formulate a 1-variable unconstrained NLP to choose a time between pavement renewals to minimize average total daily cost per mile. Evaluate any calculus integrals in your objective.