I used Mathcad:

Given (which is a keyword in Mathcad)

sqrt(y^2 + 24^2) + sqrt(x^2 + 24^2) = 120
sqrt( (y+24)^2 + (x+24)^2 ) = 120

Find (y,x)=

92.125, 6.251 which means the ladder touches the wall at
116.125 inches height (and comes out 30.251 inches from the wall).

and also the interchanged solution which you can tell by the drawing is
wrong, and two negative solutions which are just theoretical, since
Mathcad wouldn't accept x0, y0 as constraints for some reason.

with y being the distance from the top edge of the cube to where the
ladder touches, and x being the distance from the right edge of the
right edge to the ladder.

Doug