Powered by Koffee $($PBK$)$ is a new campus coffee store. PBK uses $40$ bags of whole bean coffee every month, and you may assume that demand is perfectly steady throughout the year.

PBK has signed a yearlong contract to purchase its coffee from a local supplier, Phish Roasters, for a price of $$35$ per bag and an $$75$ fixed cost for every delivery independent of the order size. The holding cost due to storage is $$1$ per bag per month. PBK managers figure their cost of capital is approximately $2$ percent per month.

$($a$)$ What is the optimal order size, in bags? $($Round the answer to $2$ decimal places.$)$

$($b$)$ Given your answer in $($a$),$ how many times a year does PBK place orders? $($Do not round intermediate calculations. Round the answer to the nearest whole number.$)$