Will they meet?

The diagram below indicates the space of outcomes. The central diagonal line indicates the meeting of A and B. The green extensions to the right and upward correspond to A and B successfully waiting 15 minutes, respectively. We must find out what fraction of the square is green.

The easiest way is to subtract the areas of the white triangles from that of the square. Each triangle is $\tfrac34$ long and $\tfrac34$ wide. Thus $\text{area of triangles} = 2\cdot \tfrac12\cdot\tfrac34\cdot\tfrac34 = \tfrac9{16},$ so that $\text{green area} = 1 - \tfrac9{16} = \tfrac7{16} = \boxed{0.4375}.$