Ladder Sliding Down Wall

Calculus 1 · Axiom Academy

A 10-ft ladder leans on a wall. Slide its foot out and the top has no choice but to slide down — at a speed the geometry sets for you. That's related rates. The bottom of a 10-ft ladder slides away from the wall at a steady 2 ft/s . The top must slide down to keep the ladder the same length — but how fast , and does it stay steady? Three moves with one relationship answer it. Slide the foot — watch the top fall Drag the foot out from the wall. The ladder length never changes, so as the bottom moves out the top is dragged down — and the further out you go, the faster the top whips down. Stop at the famous moment: bottom 6 ft from the wall. Rearrange — set the foot's speed, read the top's Freeze the moment the foot is 6 ft out (so y = 8 ft). Differentiating x² + y² = 100 gives x(dx/dt) + y(dy/dt) = 0, which solves to dy/dt = −x(dx/dt) ÷ y. Drag the foot's speed and watch the top's speed fall straight out of it. The moment matters — the answer isn't one number The foot moves out at a steady 2 ft/s, but the top's speed is anything but steady. Drag the foot and watch the top-speed bar: it crawls when the ladder stands tall, then runs away as the ladder flattens — and once the foot passes about 7.1 ft, the top falls faster than the foot slides.

This is the written version of the interactive lesson above. See the full Calculus 1 course.