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. However, the same number of checkpoints are not

necessarily in operation at all times. This is because each checkpoint requires a team of

three TSA officers to staff it$,$ and it is not always necessary to have the same number of

checkpoints in operation at any one time. Passengers arriving at the airport are first greeted

by a "greeter" who directs each passenger to a specific 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 four minutes for each pass$5.$ What is the minimum number of checkpoints at which the average passenger screening time, including wait time at checkpoint screening, is less than equal to $9$ minutes $($W between $5$:$00$ a$.$m$.$ and $6$:$00$ a$.$m$.?(5$ pts$)$

$6.$ What is the average passenger screening time, including wait time at checkpoints $($W: min$),$ at the minimum number of checkpoints that ensures that the average passenger screening time is less than equal to $9$ minutes $($W between $5$:$00$ a$.$m$.$ and $6$:$00$ a$.$m$.?(2\mathrm{.}5$ pts$)$

$7.$ What is the minimum number of checkpoints at which the average passenger screening time, including wait time at checkpoint screening, is less than equal to $9$ minutes $($W between $6$:$00$ a$.$m$.$ and $8$:$00$ a$.$m$.?(5$ pts$)$

$8.$ What is the average passenger screening time, including wait time at checkpoints $($W: min$),$ at the minimum number of checkpoints that ensures that the average passenger screening time is less than equal to $9$ minutes $($W between $6$:$00$ a$.$m$.$ and $8$:$00$ a$.$m$.?$

$(2\mathrm{.}5$ pts$)$