Question $1$:

A nursing home employs attendants who are needed around the clock. Each attendant is paid the same

regardless of when his or her shift begins. Each shift is $8$ consecutive hours. Shifts begin at $6$ A$.$M$.,10$ A$.$M$.,2$ P$.$M$.,6$ P$.$M$.,10$ P$.$M$.,$ and $2$ A$.$M$.$ The following table shows the nursing home's requirements for the numbers of attendants to be on duty during specific time periods.

a$)$ What is the minimum number of attendants needed to satisfy the nursing home's requirements?

$($b$)$ The nursing home would like to use the same number of attendants determined in part $($a$)$ but would now like to minimize the total salary paid. Attendants are paid $$16$ per hour during $8$ A$.$M$.-8$ P$.$M$.,$ and a $5\%$ premium per hour during $8$ P$.$M$.-8$ A$.$M$.$

Shift Time Number of Attendants

A $26$ AM $8$

B $610$ AM $27$

C $10$ AM $2$ PM $12$

D $26$ PM $23$

E $610$ PM $29$

F $10$ PM $2$ AM $23$