Question $3$: $[12$ marks$]$

Lingnan Barkey $($LB$)$ bakes and sells egg tarts which is a very popular product. Historically, LB has observed that the hourly demand of egg tarts is normally distributed with a mean of $200$ egg tarts and a standard deviation of $15$ egg tarts. The baking time of egg tarts $($lead time$)$ appears normally distributed with a mean of $45$ minutes and a standard deviation of $6$ minutes. LB aims to limit the stock out risk during the baking period of a new batch of egg tarts to $5$ percent at most. When should LB start to bake a new batch of egg tarts?