The Transportation Security Administration $($TSA$)$ operates "checkpoints" to screen departing passengers at the St$.$ Paul International Airport. Each checkpoint consists of a baggage conveyor belt, a CTX scanning$/$imaging machine, and a walk$-$through metal detector through which passengers pass. Each checkpoint requires a team of three TSA officers to staff it$,$ and there is not always a need to have the same number of checkpoints operating simultaneously. Passengers arriving at the airport are first greeted by a "greeter" who directs each passenger to a particular checkpoint that is in operation. Once "greeted," each passenger then joins the single line at their assigned checkpoint to be screened, and it takes an average of $3$ minutes to process each passenger through the checkpoint. The greeter's job is to evenly distribute the arriving passengers among the operating checkpoints, although the rate at which passengers arrive at the airport and approach the greeter varies with the time of day:

Time Average Number of Passengers Arriving per Hour

$5$:$00$ AM to $6$:$00$ AM $120$

$6$:$00$ AM to $8$:$00$ AM $150$

Recommend the minimum number of checkpoints that TSA should operate during the various time intervals of the day first described in the table above, keeping in mind that TSA would prefer to minimize staffing requirements based on the number of checkpoints $($use five decimal places$).$

$1.$ What is the minimum number of checkpoints required from $5$:$00$ a$.$m$.$ to $6$:$00$ a$.$m$.(2$ pts$)?$

$2.$ What is the average wait time $($min$)$ based on the minimum number of checkpoints required from $5$:$00$ AM to $6$:$00$ AM $(3$ pts$)?$

$3.$ What is the minimum number of checkpoints required from $6$:$00$ a$.$m$.$ to $8$:$00$ a$.$m$.(2$ pts$)?$