A manufacturer has a steady annual demand for $16,128$ cases of sugar. It costs $$7$ to store $1$ case for $1$ year, $$32$ in set up cost to produce each batch, and $$18$ to produce each case.

a$)$ How many cases of sugar should be produced to minimize inventory costs?

b$)$ The manufacturer should manufacture batches of sugar annually.

$($Type a whole number.$)$

c$)$ What are the total inventory costs for the year? $