Problem $5(15$ points$).$ A manufacturing plant procures $100,000$ filters per year. Each unit has a fixed order cost is $$50.$ To maintain the inventory in optimal conditions, there is a carrying cost rate of $10\%$ per year. The vendor of filters has offered the following price breaks: for orders of more than $5,000$ filters, cost is $$4\mathrm{.}50/$unit; for orders between $1,000$ and $5,000$ filters, cost is $$4\mathrm{.}75/$unit; orders less than $1,000$ filters have a cost of $$5\mathrm{.}00/$unit$.$

$(8$ points$).$ Find the economic order quantity.

$(7$ points$).$ Calculate the average annual cost for the optimal solution.