Eric and Joyce are the first two to decide to take an answer to the guard. They're both wrong. It appears that Eric guessed "aquarium" and Joyce guessed "planetarium." So they're apparently just randomly mixing up letters to form any words at all that seem to spell the names of places, and that seems... very unlikely to work. As they return and everyone starts milling around again, Dustin looks up at the walls and notices the pictures, so she figures the puzzle out pretty quick after that. There was kind of a perception that Dustin figured this out immediately upon entering the room, like she was some kind of a genius, but it took her at least a little while, so I think it's more that she figured it out before the rest of the yahoos. Dustin connects the letters on her sheet to a place called "Chuquicamata." She leaves and gives her answer to the guard, and she's correct. She and Kandice get the clue, which tells them to fly to Calama, Chile, and go to that very copper mine, Chuquicamata. It's actually the world's largest open-pit copper mine, so that should be fun and not at all environmentally hazardous. In the cab, Dustin calls the thing "frickin' hard," so again, I don't think she got it immediately, savant-style. She just hopes people didn't see her reading it off the picture on the wall. Apropos of nothing, I'm interested in the fact that Kandice has a Nike swoosh on her baseball cap, and I recall Esquire Rob having a Nike swoosh on a shirt that had to be covered up with tape. Maybe they've gotten more lax, or maybe Nike likes being associated with Beauty Queens better than with That Guy.
It's not clear whether Eric did or didn't see Dustin, but he has finally noticed the pictures. However, when he looks at his list of letters, he sees no "H," and there's an "H" in the name "Chuquicamata," so he discards that as a possibility, which buys him a ticket to a world of hurt. He has unfortunately missed the "H" that's on a guy's pen at the big boardroom table.