A sketch of the scenario in the question above gives a right triangle with one of the angles as 45 degrees and the side adjacent to the angles as 15 feet. The length of the ladder forms the hypotenuse of the right triangle.
Recall that the ratio of the two legs and the of a 45 degrees right angle is [tex] 1:1:\sqrt{2} [/tex].
Let the length of the ladder be L, then
[tex] \frac{1}{\sqrt{2}} =\frac{15}{L} \\ \\ \Rightarrow L=15\sqrt{2}\\ \\ =21.2\ feet [/tex]
