A toy manufacturer uses $52,800$ rubber wheels per year for its popular dump truck series. The firm makes its own wheels, which it can produce at a rate of $800$ per day. The toy trucks are assembled uniformly over the entire year. Carrying cost is $$1\mathrm{.}80$ per wheel per year. Setup cost for a production run is $$47.$ The firm operates $240$ days per year. Determine the following:

a$.$ Optimal run size $($Round your answer to a whole number, following normal rules of rounding.$)$

b$.$ Use your final answer from part a to determine minimum total annual cost for carrying and setup. $($Round your answer to a whole number.$)$

c$.$ Cycle time for the optimal run size $($Round your answer to two decimal points.$)$

d$.$ Run time $($Round your answer to two decimal points.$)$