Related Rates

Calculus 1 · Axiom Academy

When two quantities are tied together by an equation, their rates of change are tied together too — differentiate the link, and one rate gives you the other. 1. Linked Quantities, Linked Rates Pump air into a spherical balloon at a constant rate. Its volume V and radius r are locked together by , so as one moves the other must move with it. But watch the speeds: even though air enters at a steady 10 cm³/s, the radius grows fast at first and then slows down . The rates are linked — they are not equal. 2. Differentiate the Link with Respect to Time Take of both sides of . Because r depends on t , the chain rule attaches a to the right side — that factor is what turns an equation between quantities into an equation between rates. differentiate both sides with respect to t The factor is exactly — the slope of the V – r curve . The animation shows that slope getting steeper as r grows: a steeper slope means each bit of radius adds more volume, so to keep fixed the radius must change ever more slowly. That is the slowdown from Step 1, made precise. 3. Substitute the Known Values, Then Solve Only after differentiating do the specific numbers go in. For the balloon we are given cm³/s and we want at the instant r = 5 cm. Substitute both into the rates equation, then solve. The radius is increasing at cm/s when r = 5 cm. The units, , confirm a length-per-time rate — a built-in sanity check. A second instance: the sliding ladder

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