You have been asked to analyze the queue for registrations for a popular Financial Seminar at a major national conference.$$ Conference attendees arrive to register at the rate of $3$ people every $12$ minutes $($Poisson distributed$).$ It takes each host, any number of which can be hired directly through the conference at the cost of $$15$ per hour. It takes $6$ minutes to register a guest, following a $($negative exponential distribution$).$ The conference recognizes a waiting cost of $$55$ per hour per guest in line. $($The goal is to not have the attendees wait in line more than $6\mathrm{.}5$ minutes$).$

A single waiting line is set up$,$ but can accommodate multiple servers.$$

a$.$ Based on the goal stated above, what is the minimum number of hosts that could be hired for this system?$[$ Select $]["1","4","3","2","5"]$ hosts

b$.$ Based on wait cost, what is the optimal number of servers needed to minimize total costs?$[$ Select $]["1","4","5","2","3"]$ hosts

c$.$ Rounded to the nearest dollar, what is total per day $($wait$)$ cost at the optimal number of servers if the registration desk is open for $3$ hours? $[$ Select $]["$$$203\mathrm{.}31","$$$58\mathrm{.}02","$$$174\mathrm{.}08","$$$62\mathrm{.}46","$$$342\mathrm{.}89"]$

d$.$ At the optimal number of servers, what is the average time to check in an attendee? $[$ Select $]["1\mathrm{.}11$ minutes", $"\mathrm{.}55$ of a minute", $"\mathrm{.}94$ of a minute", $"8\mathrm{.}12$ minutes", $"6\mathrm{.}94$ minutes"$]$

$$