20k/120 needs to be rounded down to 166 patrons because having 167 patrons for every 120 ft^2 implies that the room must be at least 20,040ft^2. As that cannot be the case, we need the room to hold a maximum of 166 patrons. Should Queen Bey be included in the 166 individuals, or should she be the 167th person? idk. The problem isn’t clear enough.

Furthermore, 141ft^2 of the entire establishment would not be correct because the square root of 20k ft^2 is measured in ft, not ft^2. “Beyonce’s voice only travels the square root of the room size without any equipment and the room size is 20,000 square feet” implies her voice is audible up to approximately 141.4ft away. Assuming that Beyoncé can be heard from just over 141.4ft away in ANY direction entails that anyone located in a circular region pi times larger than 20kft^2 will be able to hear her. Note that Beyoncé must be standing at the center of this region.

In order to know how many of the single ladies in the club will raise their hands, we need to know the shape of the room, its actual dimensions, where Beyoncé is standing, and where the patrons are standing. Soooo many questions need to be answered. Are the patrons evenly spaced out or bunched up into groups (this seems most plausible)? If each person does have 120ft^2 to themselves(which seems unlikely), where are they standing/dancing within that 120ft^2? What shapes are each of those 120ft^2 “pieces” of dance floor made up of? Is Beyoncé in the middle of the room? In a corner? Against a wall? The initial problem is too ambiguous for us to know.

One thing is clear: If the room is shaped like a disk, all the single ladies will put their hands up.

As an aside, if the club is shaped like a square, I propose a new mathematical concept: the “All the Single Ladies” Set. That is, If Beyoncé stands at any point within the “All the Single Ladies” Set, all the single ladies in “the club” will put their hands up.

 Furthermore, any single lady standing within the “All the Single Ladies” set will hear the Queen’s call regardless of where Bey is located in “the club.” That is, the single ladies standing within “All the Single Ladies” are guaranteed to always put their hands up when the Queen commands.