The $21,000$ seat air E arena houses the local professional ice hockey basketball indoor soccer and arena football teams as well as various trade shows wrestling and boxing matches tractor pools and circuses a ring of vending annually sells large quantity of soft drinks and beer and plastic cups with the name of the arena and the various team logos on them the local container cup manufacture that supplies the cups and boxes of $100$ has offered arena management the following discount prices scheduled for cups $2000$ to $6900$ and $$9947$ a box $7000$ to $11,999$ $$43$ per box $12,000$ to $19,999$ forty $$1\mathrm{.}00$ a box $20,000+38$ dollars a bucks the annual demand for cups is $2\mathrm{.}3$ million the annual carrier cost per box of cups is $$1\mathrm{.}90$ an ordering cost is $$320$ determine the optimal order quantity and total annual inventory cost