The Expanding Ripple

Calculus 1 · Axiom Academy

Drop a pebble and the ripple's radius grows steadily — but its area races ahead. When one thing changes, how fast does everything tied to it change? A pebble hits a still pond and a circular ripple races outward. Its radius grows at a steady pace. But area depends on the square of the radius, so the area doesn't grow steadily at all — it speeds up. That mismatch between two linked speeds is exactly what related rates is about. Press play and watch the clock run. The radius climbs at a constant dr/dt = 10 cm/s , yet the area readout climbs faster and faster — because every second the ripple is bigger, the same outward step paints a wider band of new water. Radius rises in a straight line; area curves upward. The area's rate dA/dt isn't constant — it grows right along with the radius. Dial in the radius and the radius-speed Here is the ripple frozen at one instant. Set how fast the radius is expanding (dr/dt) and how big the ripple currently is (r), and read off the area's speed dA/dt straight from the formula. Push r up at a fixed dr/dt and watch dA/dt climb — the area-rate depends on where the rim is right now. dA/dt = 2πr·(dr/dt). Same dr/dt, bigger r → bigger dA/dt. The area's speed is set by how far out the rim already is.

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