Consider a device that follows the Weibull lifetime distribution with n $=8000$ hours and $=2.$ If it experiences a failure, minimal repair will be performed. You are to determine the optimal preventive replacement age of this device by minimizing the average unit time cost. Suppose that preventive replacement cost $=$ $$1000,$ preventive replacement time $=4$ days, minimal repair time $=1$ day, minimal repair cost $=$ $$300,$ and there are $8$ hours in a day. Feel free to use Excel or Matlab. $(1)$ What is the optimal preventive replacement age of this device by minimizing the average unit time cost? $(2)$ What is the expected total cost at the optimal preventive replacement age? $(3)$ What is the minimal average unit time cost?