ANSWER COMPLETELY AND SHOW ALL WORK

The maintenance department of a large hospital uses about $816$ cases of liquid cleanser

annually. Ordering cost are $$12,$ carrying costs are $$4$ per case a year, and the new price schedule

indicates that orders less than $50$ cases will cost $$20$ per case, $50$ to $79$ cases will cost $$18$ per

case, $80$ to $99$ cases will cost $$17$ per case, and larger orders will cost $$16$ per case. Determine

the optimal order quantity and total cost.