Leach Finishing makes various metal fittings for the construction industry. Three of the fittings, models X$-12,$ X$-24,$ and X$-30,$ require grinding on a patented machine of which Leach has only one. The cost of production information for the three products follow:

X$-12$ X$-24$ X$-30$

Price per fitting $ $31$ $ $47$ $ $66$

Variable cost per fitting $ $16$ $ $23$ $ $40$

Units per hour of grinding $18\mathrm{.}012\mathrm{.}510\mathrm{.}0$

The testing machine used for both models has a capacity of $2,990$ hours annually. Fixed manufacturing costs are $$488,000$ annually.

Required:

a$.$ Suppose that Leach Finishing can sell at most $56,800$ units of any one fitting. How many units of each fitting model should Leach Finishing produce annually?

b$.$ Suppose that Leach Finishing can sell at most $16,200$ units of any one fitting. How many units of each fitting should Leach Finishing produce annually?