The Pacific Lumber Company and Mill has demand for and processes $10,000$ logs annually $($this is the amount they sell to their customers$),$ operating $250$ days per year. Immediately upon receiving an order, the logging company$$s supplier begins delivery to the lumber mill, at a rate of $60$ logs per day until the whole order is received. The lumber mill has determined that the ordering cost is $$1,600$ per order and the cost of carrying logs in inventory before they are processed is $$15$ per log on an annual basis. Determine the following:

What is the total inventory cost associated with the optimal order quantity?