Related Rates

Calculus 1 · Axiom Academy

When two quantities are tied by an equation, their rates of change are tied too. Blow up a balloon and the radius grows — but so does the volume, and so does the surface area. None of those is free to wander on its own: they are locked together by geometry. So the speeds they change at must be locked together too. That linkage is what related rates is about. Watch the balloon inflate at a steady radius rate. The radius ticks up evenly, yet the volume rate races ahead — because a fatter balloon needs far more air per millimeter of radius. The two rates are different numbers, but they move in lockstep. Same clock, two speeds: a constant radius rate drives an ever-growing volume rate. Next, you will set those numbers yourself. You control how fast you pump — the radius rate . The balloon decides the rest. Pick a pumping speed and a current radius; the volume rate falls right out. Notice how much it jumps as the balloon gets bigger. At the volume swells by about — over a hundred times the radius rate. Same pump, wildly different rate, set entirely by the size right now. Where the link comes from: differentiate the relation Both quantities obey one equation, . Differentiate both sides with respect to time and the chain rule hands you the link. Geometrically: nudge the radius out by a hair and you wrap the balloon in a thin shell — its volume is the surface area times the thickness, . Drag the radius and watch that factor appear.

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