A toy manufacturer uses $47,600$ rubber wheels per year for its popular dump truck series. The firm makes its own wheels, which it can produce at a rate of $900$ per day. The toy trucks are assembled uniformly over the entire year. Carrying cost is $$1\mathrm{.}20$ per wheel per year. Setup cost for a production run is $$40.$ The firm operates $238$ 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.$)$