The Clark County Sheriff's Department schedules police officers for $8-$hour shifts. The beginning times for the shifts are $8$:$00$ a$.$m$.,$ noon, $4$:$00$ p$.$m$.,8$:$00$ p$.$m$.,$ midnight, and $4$:$00$ a$.$m$.$ An officer beginning a shift at one of these times works for the next $8$ hours. During normal weekday operations, the number of officers needed varies depending on the time of day. The department staffing guidelines require the following minimum number of officers on duty.

Time of Day Minimum No$.$ of Officers on Duty

$8\mathrm{.}00$ a$.$m$.$noon $8$

Noon$4$:$00$ p$.$m$.9$

$4$:$00$ p$.$m$.8$:$00$ p$.$m$.13$

$8$:$00$ p$.$m$.$midnight $10$

Midnight$4$:$00$ a$.$m$.7$

$4$:$00$ a$.$m$.8$:$00$ a$.$m$.9$

Determine the number of police officers who should be scheduled to begin the $8-$hour shifts at each of the six times to minimize the total number of officers required. $($Hint: Let

x$1=$

the number of officers beginning work at $8$:$00$ a$.$m$.,$

x$2=$

the number of officers beginning work at noon, and so on$.)$

$($x$1,$ x$2,$ x$3,$ x$4,$ x$5,$ x$6)=$