$(15$ points$)$ Suppose that a metal rod is supposed to be cut $42\mathrm{.}5mm.$ in length $($target$).$ The upper specification limit is set at $42\mathrm{.}9mm.,$ and the lower specification limit is set at $42\mathrm{.}1mm.$ A metal rod's distance that is below the lower specification limit must be scrapped at a cost of $$12\mathrm{.}00$ while a metal rod's distance that is above the upper specification limit must be scrapped at a cost of $$15\mathrm{.}00$ since longer rods cost more to make. The company uses the standard Taguchi Quality Loss Piecewise Function to estimate quality costs. Suppose that a sample of $5$ units was taken, and the rod measurements were $42\mathrm{.}3,42\mathrm{.}15,42\mathrm{.}8,42\mathrm{.}5,$ and $43\mathrm{.}4mm.,$ respectively.

a$.$ First draw the graph of the Taguchi quality loss function and write down the entire piecewise function $($Recall that a piece wise function will consist of multiple functions$).$

b$.$ What is the Taguchi Quality Cost of that sample of $5$ units? Hint: Calculate the quality deviation costs for each rod and then sum these costs together?

c$.$ Suppose that a new process is being considered. In a test, a sample of $5$ rods yielded lengths of $42\mathrm{.}53,42\mathrm{.}4,42\mathrm{.}6,42\mathrm{.}62,$ and $42\mathrm{.}37mm.,$ respectively. Based on your comparative samples, if the new process would cost $$2\mathrm{.}00$ per part to implement, should it be implemented? Justify your answer.