$($Process analysis $30$ pts$.)$

To prepare for a festival's potluck, Griselda decides to make homemade falafel. After a grueling three hours, she manages to fry a total of $270$ falafels.

a$)$ Her pot size currently fries $20$ falafels at a time. How long is each falafel fried for? Hint think of the pot as the amount of inventory in our system.

b$)$ Griselda is considering purchasing a larger pot that can fry as many falafels as she wants per batch. To cook them to perfection, each batch of falafels is fried for $12$ minutes. If she wants to continue producing falafels at her current rate of $270$ per $3$ hours, how many falafels on average should be fried per batch? Should she purchase a larger pot?