The "Ladder Problem" is a classic exercise in calculus.

A ladder with length 5 units leans against a wall (represented by the vertical axis below.) The bottom of the ladder slides across the floor while the top of the ladder slides along the wall.

At what rate does the point B move ? and what rate does the point C move ? Does this rate depend on the relative distances of the points from A?

To answer the question, we can let x equal the distance from point A to C in the diagram below.
And let y equal the distance from point A to B. Then by the Pythagorean theorem, x^2 + y^2 = 5^2.
Using implicit differentiation, with respect to t, we find dy/dt = -(x/y)(dx/dt).
What this means is the following (assuming dx/dt is constant):
- when y decreases, the change in y with respect to time is increasing (you can see B move faster when it gets lower.)
- when x decreases, the change in y with respect to time is decreasing also (B slows when C approaches A.)

Try moving point C below to develop an intuitive sense of the problem.

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The Ladder Problem


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