Analytic Essay: is a brief work of nonfiction in which a writer explores a subject by breaking it into parts. ... I think that "The American Idea" is both an analytic essay and an interpretive essay. This is because in an analytic essay he explores how America developed and how people came to make a better life.
It makes sense. I would say that it means that her cheeks are blushing, but it could also mean that she is embarrassed. Those two are linked, as people blush when embarrassed.