Problem

A ladder 10 feet long is resting against a wall. If the bottom of the ladder is sliding away from the wall at a rate of 1 foot per second, how fast is the top of the ladder moving down when the bottom of the ladder is 8 feet from the wall?

Solution

The picture looks like this.

We have dx/dt = 1 ft/sec, and we want dy/dt.

x and y are related by the Pythagorean Theorem

Differentiate both sides of this equation with respect to t to get

When x = 8 ft, we have

therefore

The top of the ladder is sliding down (because of the negative sign in the result) at a rate of 4/3 feet per second.