A ladder 10 feet long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 feet from the wall?

Core Concept

Let x be the distance from the wall to the bottom of the ladder, and y be the height of the top of the ladder on the wall. By the Pythagorean theorem: x² + y² = 100. Differentiating with respect to time: 2x(dx/dt) + 2y(dy/dt) = 0. When x = 6, y = 8. Given dx/dt = 1 ft/s: 2(6)(1) + 2(8)(dy/dt) = 0 → 12 + 16(dy/dt) = 0 → dy/dt = -12/16 = -0.75 ft/s.