the audience rather than a character alone
The correct answer is After I took a nice hot shower, my cat lapped up the water on the floor.
Explanation:
A dangling modifier is a term used for an ambiguous grammatical construction in which one word might be incorrectly linked to a modifier (a word that modify a noun). In the example given, in the case given the relationship between nouns and verbs is not clear as it is not clear whether it was the cat who took a hot shower and then lapped up the water or there is another subject involved. However, in the sentence "After I took a nice hot shower, my cat lapped up the water on the floor" each of the actions in the sentence (took a shower and lapped up) are linked to a subject in a clear way so there is no ambiguity in which subject carried out each action and thus the dangling modifier disappears.
A. or "The family is letting us stay at their cottage"
The other one makes no sense in proper grammar