Question $6(20$ points$)$

Rhythm Revolution Dance Studio has just launched in the city center. The studio has generated buzz so swiftly that applications are flooding in from aspiring dancers all over the globe, eager to enroll in classes. The studio started accepting applications $6$ months ago, and the data below shows the number of applications received each month since then:

$\backslash$table$[[$Month$,$Applications during that month$],[1,700],[2,2200],[3,1800],[4,500],[5,400],[6,300]]$

Studio manager, Max, leads a team of $12$ admissions consultants, each of whom works $20$ days per month. Each consultant can evaluate $5$ applications per day.

$($i$)$ What is the capacity in applications$/$month for the entire team?

$($ii$)$ What is the average input rate, $Ri(t)$ for all the applications coming in over the $6$ months?

$($iii$)$ Characterize the output rate over time, $Ro(t)$ for these applications

$($iv$)$ Is this process stable?