Write Every Moving Letter as a Function of Time, Then Differentiate

You'll learnto build the relationship a related-rates problem actually needs, and to apply d/dt to it so that every varying letter picks up its own rate factor instead of quietly losing one.

A related rates problem hands you one rate and asks for another. The arithmetic is rarely the difficulty. Almost everything that goes wrong here goes wrong before a single number is substituted: the wrong relationship gets written down, something that is moving gets treated as fixed, or the chain rule quietly fails to happen. Watch the slick. Its radius grows at a steady 0.5 metres per second and never speeds up — yet the area races away, because the d/dt operator multiplies that one measured rate by 2πr, and 2πr is the rim, which is getting longer.

8 STEPS · 6 QUICK CHECKS · EVERY MOVING LETTER GETS A RATE FACTOR · A CONSTRAINT REMOVES THE RATE YOU DO NOT HAVE · v1

A(t) = π [ r(t) ]² ⇒ dA/dt = 2π r · dr/dt · one measured rate drives another

Before you start

What you're looking at

An oil slick spreading in a circle. Its radius r(t) grows at a constant 0.5 metres per second, which is the one rate anybody measured. The pale ring around it is the area the slick will add in the next 0.6 seconds.

The question

The radius rate never changes. So why does the area rate keep climbing — and what exactly does d/dt do to A = πr² to tell you that?

Watch for

The moment the factor dr/dt arrives on screen. It is the derivative of the inside function, and dropping it leaves an equation whose two sides are not even the same kind of quantity.

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