A process that makes plastic cell phone cases uses an injection molding machine to create the cases. The target thickness of the cases is set by a controller on the machine, and is currently set to deliver a case with a thickness of $5\mathrm{.}1mm.$ Three cases are pulled each hour, and the thickness of those cases is measured. The machine has been operating smoothly over the last month, and the data from the samples have a sample mean of $5\mathrm{.}10mm,$ and a sample standard deviation of $\mathrm{.}04mm.$

The specs for the thickness of the cell phone case have been set at $4\mathrm{.}7mm-5\mathrm{.}5mm.$

a$)$ Is this process capable?

b$)$What target should the machine be set for to ensure capability while still using the least amount of plastic?

c$)$ If plastic currently represents an annual cost to the company of $$10,000,000,$ approximately how much money will the company save by making this change?