6. An equipment planner of a job shop intends to keep a particular lathe for a years. From past data, he knows the lifetime of the lathe follows an exponential distribution with a mean of 5 years. If the lathe fails before x, he must replace it by a new one, which costs c dollars. If the lathe survives to time x, he can trade it in for a new one at c/2 dollars. What is the planner's average cost per year, for a given x? What is the optimal value of r that minimizes this cost? Will this optimal value change if the lifetime of the lathe follows a non-exponential distribution; why