A manufacturer has an annual demand for $14,112$ cases of sugar. It costs $$$$4$ to store a case for $1$ year, $$$$36$ in setup cost to produce each$$ batch, and each case costs $$14$ to produce.

a$.)(4$ pts$)$ How many cases should they per batch to minimize total costs?

b$.)(2$pts$)$ How many batches will they make each year?