A company is about to begin production of a new product. The manager of a department that is asked to produce one of the components wants to know if there is enough machine time available. The machine will produce the item at a rate of $360$ units a day. Ninety units will be used daily in assembling the final product. The company operates five days a week, $50$ weeks a year. The manager estimates that it will take almost a full day to get the machine ready for a production run, at a cost of $$260.$ Inventory holding cost will be $$13$ per unit per year.

a$.$ What production run quantity should be used to minimize total annual setup and holding cost? $($Do not round intermediate calculations. Round the final answer to the nearest whole number.$)$

Production run quantity

units

b$.$ What is the length of a production run $($in days$)?($Do not round intermediate calculations. Round the final answer to $2$ decimal places.$)$

Run length

days

c$.$ During production, at what rate will inventory build up$?$

Rate

units

d$.$ If the manager needs to run another job between runs of this job, and needs a minimum of $7$ days per cycle of this job for the other job, will there be enough time?