A ladder rests against a wall with its lower end at a distance x from the wall and its upper end at a height 2x above the floor. If the lower end slides through a distance y perpendicular to the edge between the floor and the wall, by how much does the upper end slide?

The initial length of the ladder is the hypotenuse of a right triangle with base x and height 2x, which equals sqrt(x^2 + (2x)^2) = x sqrt(5). After the lower end slides by y perpendicular to the wall, its new distance from the wall becomes (x + y). If the new height is h, applying the Pythagorean theorem gives h = sqrt(L^2 - (x + y)^2) = sqrt(5x^2 - (x + y)^2). The distance the upper end slides down is the difference between the initial and new heights, yielding 2x - sqrt(5x^2 - (x + y)^2).

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