The same production machine is used to produce a batch of parts that each has a starting material cost of $C_m=$$$2\mathrm{.}5.$ Batch quantity $Q=150.$ The actual processing time in the operation $T_o=3\mathrm{.}25$ min. Time to load and unload each workpiece $T_h=1\mathrm{.}5$min. Tool cost $=$$$4\mathrm{.}0,$ and each tool can be used for $20$ pieces before it is changed, which takes $2\mathrm{.}0$min. Before production can begin, the machine must be set up$,$ which takes $T_{su}=1\mathrm{.}5$ hours. Hourly wage rate of the operator $R_H=$$$\frac{10\mathrm{.}50}{h}r,$ and the applicable labor overhead rate $R_{\text{L}\text{O}\text{H}}=35\%.$ Determine the:

$($a$)$ cycle time $T_c$ for the piece,

$($b$)$ avg. production rate $R_p$ when setup time is figured in$,$ and

$($c$)$ cost per piece $C_{pc}.$