Related Rates Calculator — Sliding Ladder Problem
A ladder of fixed length L leans against a wall. Its base slides away from the wall at a known rate while the top slides down. Because x² + y² = L² at every instant, differentiating with respect to time links the two rates: 2x(dx/dt) + 2y(dy/dt) = 0. Enter the ladder length, the current base distance, and the base's rate to get the ladder's height and how fast the top is sliding down.
Setting Up the Sliding Ladder Equation
A ladder of length L leans against a wall, base x feet out and top y feet up. Since x² + y² = L² at every instant, differentiating with respect to time t gives 2x(dx/dt) + 2y(dy/dt) = 0, which rearranges to dy/dt = −(x/y)·(dx/dt) — the top's rate depends on the current position and the base's rate.
Worked Example: A 10-Foot Ladder
A 10-ft ladder has its base sliding away from the wall at 2 ft/s. When the base is 6 ft out, the top's height is y = √(10² − 6²) = √64 = 8 ft. The rate it slides down is dy/dt = −(6/8)×2 = −1.5 ft/s — the top drops a foot and a half per second, even though the base only moves at 2 ft/s.
Why the Top Accelerates Near the Ground
As the base nears the point directly under the top (x approaching L), y shrinks toward 0 and x/y grows without bound, so dy/dt grows toward negative infinity even at constant dx/dt. A ladder sliding at a steady pace looks like it crashes down suddenly at the end — geometry, not extra effort, drives it.
Frequently Asked Questions
What is a related rates problem?
A related rates problem links two quantities that change together through a fixed equation - here x² + y² = L² for a ladder against a wall - then uses calculus to find how fast one changes when you know how fast the other does.
Why does the calculator show dy/dt as negative?
Negative means the top is sliding down, since y decreases as x increases. If you only need the speed, drop the sign; if you need direction too, negative means downward.
Does this calculator handle other related-rates scenarios, like a cone filling with water or a growing shadow?
No - this tool solves only the sliding ladder case (x² + y² = L²). Cone, shadow, and other related-rates problems start from a different equation and need a dedicated setup.
What happens as the base approaches the ladder's own length (x → L)?
The top's height y shrinks toward 0, and since dy/dt = −(x/y)·(dx/dt), the rate grows toward negative infinity. A ladder nearly flat on the ground seems to drop its last few inches almost instantly.
What units should I use?
Any consistent length and time units - feet and seconds, meters and seconds, and so on - as long as the ladder length and base distance share the same length unit and the rate matches both.