Animate classic related rates problems: expanding balloon, sliding ladder, filling cone. See rates change in real time.
The three textbook setups come to life here: a spherical balloon inflating at a set volume rate, a ladder sliding out from a wall, and a cone filling with water. You set the dimensions and the given rate; the tool moves the picture accordingly and shows every other rate at the current moment. The surprise for most people: none of the derived rates are constant, even when the rate you set is. A ladder whose bottom slides at a steady 1 ft/s drops its top faster and faster, because the rate depends on the changing geometry, not just on the input number.
A concrete input and expected output from the current implementation.
Input
Ladder: length 10 ft, bottom slides away from the wall at 1 ft/s. Pause the animation when the bottom is 6 ft from the wall.
Expected output
At that moment the top is sliding down at 0.75 ft/s; the readout shows dy/dt = -0.75 ft/s, negative because the top descends.
With x the bottom distance and y the top height, x^2 + y^2 = 100. Differentiating with respect to time gives 2x(dx/dt) + 2y(dy/dt) = 0, so dy/dt = -(x/y)(dx/dt) = -(6/8)(1) = -0.75 ft/s.
Each scene is one geometric relation: the Pythagorean relation for the ladder, the sphere volume formula for the balloon, and the cone volume formula with the water surface locked to the cone's fixed height-to-radius ratio. The displayed rates are the chain-rule derivatives of that relation; for the cone, depth and surface radius both change together, in the proportion the cone's shape fixes.
The readouts update continuously as the animation runs, and they are instantaneous rates at the moment shown, not averages over the run. Each has a sign: negative means the quantity is shrinking, as when the ladder top slides down. Pause at the geometry you care about before comparing with a hand calculation, because the numbers at two different moments differ.
Because the ratio of bottom distance to top height grows as the ladder flattens. At 6 ft out the top falls at 0.75 ft/s; at 8 ft out it falls at about 1.33 ft/s, already faster than the bottom moves. Nothing you set is accelerating; the tool is showing what the changing geometry does to the rate.
Only if your problem is one of the three built-in shapes. The tool animates the expanding balloon, the sliding ladder, and the filling cone with adjustable numbers; it does not accept an arbitrary relation of your own. If your problem is one of those shapes, changing the dimensions and the given rate covers it.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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