Related Rates
Link two changing quantities by an equation, differentiate the equation with respect to time, and solve for the rate you cannot measure directly.
What a learner can do afterwards
- Set up a relation between two quantities before differentiating anything
- Differentiate a geometric formula with respect to time
- Substitute the instantaneous values only after differentiating, and say why
1 · Read
Some problems give you two quantities that change with time. A balloon has a growing radius and a growing volume. The two rates are linked because one equation links the two quantities. Your job has four moves. Name each changing quantity with its units. Write the equation that links them. Differentiate both sides with respect to time. Substitute the instant values only after that.
A circle has radius 8 cm, growing at 3 cm per second. Area and radius obey A equals pi r squared. Differentiate with respect to time and the chain rule gives dA dt equals 2 pi r dr dt. Now substitute r equals 8 and dr dt equals 3. The area grows at 48 pi square cm per second.
A sphere obeys V equals 4 thirds pi r cubed. Differentiate first and you get dV dt equals 4 pi r squared dr dt. Only now may you substitute numbers such as r equals 3. If you substitute too early, the radius freezes into a constant and its rate silently vanishes. Finish with units, since cubic cm per second tells a different story from square cm per second.
A 10 m ladder slides down a wall. Its foot sits 6 m from the wall and slides out at 2 m per second. Foot and height obey x squared plus y squared equals 100, so y equals 8. Differentiate to get 2 x dx dt plus 2 y dy dt equals 0. Substitute and solve: dy dt equals negative 1.5 m per second. The minus sign says the top slides down.
List each quantity with its units before you start. That bookkeeping turns a confusing paragraph into a routine calculation. And when your answer comes out negative, trust it: the quantity is shrinking.
Write the link, differentiate with respect to time, then substitute the instant values.
2 · Watch
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8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.