The annual demand for Prizdol, a prescription drug manufactured and marketed by the NuFeel Company, is normally distributed with mean $50,000$ and standard deviation $12,000.$ Assume that demand during each of the next $10$ years is an independent random number from this distribution. NuFeel needs to determine how large a Prizdol plant is to build to maximize its expected profit over the next $10$ years. If the company builds a plant that can produce x units of Prizdol per year, it will cost $$16$ for each of these x units. NuFeel will produce only the amount demanded each year, and each unit of Prizdol produced will sell for $$3\mathrm{.}70.$ Each unit of Prizdol produced incurs a variable production cost of $$0\mathrm{.}20.$ It costs $$0\mathrm{.}40$ per year to operate a unit of capacity. a$.$ Among the capacity levels of $30,000,35,000,40,000,45,000,50,000,55,000,$ and $60,000$ units per year, which level maximizes expected profit? Use simulation to answer this question.

b$.$ Using the capacity from your answer to part a$,$ NuFeel can be $95\%$ certain that actual profit for the $10-$year period will be between what two values?