$.(25$ points$)$ After your success in the t$-$shirt industry, you decide to start an ice cream empire. Your

factory can produce your own ice cream at a rate of $1$ serving per second at a cost of $$1\mathrm{.}00$ per serving.

Starting a production run costs $$1200.$ Assume that the holding cost of ice cream is associated with

cooling it $($$$0\mathrm{.}01$ per serving per minute in energy costs, which is surprisingly expensive$)$ rather than

opportunity costs; ingredients can be cooled for free. Demand for ice cream is a steady $50$ servings per

minute, all of which must be fulfilled.

$($a$)$ How much ice cream should you produce in each batch, and how long will a production run for a

batch take?

$($b$)$ What should your maximum inventory level