The Sliding Ladder
Related rates · Calculus
Skill goal: Use implicit differentiation to relate how fast connected quantities change.
1 · Observe
A 10-ft ladder leans against a wall. The bottom slides out along the ground — and the top slides down the wall. The two motions are linked by the ladder’s fixed length:
x2 + y2 = 100
The wall is the y-axis, the ground is the x-axis. Drag the ladder to change its angle, or press Animate below to watch it slide.
2 · Manipulate
Tip: drag the ladder on the diagram above, or let it animate. The ladder angle is clamped between 5° and 85° so the math never divides by zero.
3 · Predict
Q1. At the instant x = 6 ft (with dx/dt = 2 ft/s), what is dy/dt?
Q2. As the ladder gets flatter (x grows, y shrinks), does the top fall faster or slower?
4 · See
Live readouts — drag the ladder or animate and watch the rates respond:
5 · Explain
The length never changes, so differentiate x² + y² = 100 with respect to time t. Each variable needs the chain rule:
2x·dx/dt + 2y·dy/dt = 0
Solve for the rate we want:
dy/dt = −xy·dx/dt
- The negative sign means y shrinks as x grows — the top falls while the bottom slides out.
- The factor x/y is a lever: when the ladder is nearly flat (y tiny), x/y is huge and the top plunges.
- This is the whole idea of related rates: differentiate the relating equation, then substitute the instant’s values.
6 · Challenge
Three rounds — each is checked automatically. Round 3 is conceptual.
Score: 0 / 3