It's mentioned that the home belongs to the family, by using 'a' the meaning changes to 'any home'. Hence option D is incorrect. When an expression is used concerning time and place 'at home', an article is not necessary. At the same time, it's specified when they'll be home at a particular time, hence the article 'the' is required before 'afternoon'. Hence option B is correct. Option A is incorrect because the indefinite article is for non specific nouns, 'afternoon' in the context of the sentence is specific. Option C is incorrect because 'at home' does not need an article whereas 'afternoon' does.