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Related rates problems find how fast one quantity changes given the rate of change of a related quantity — both changing with time. The key technique: write an equation relating the variables, differentiate both sides implicitly with respect to time t, substitute known values.
Classic examples: a spherical balloon being inflated (volume and radius related by V = (4/3)πr³ — if dV/dt is known, find dr/dt), a ladder sliding down a wall (x² + y² = L² — if dx/dt is known, find dy/dt), or water draining from a conical tank.
Step-by-step strategy: (1) Draw a diagram and label all variables. (2) Write an equation relating the variables. (3) Differentiate both sides with respect to t using the chain rule. (4) Substitute known values and solve for the unknown rate.
Worked example — ladder: A 10-ft ladder leans against a wall. The bottom slides away at 2 ft/s. How fast is the top sliding down when the bottom is 6 ft from the wall? x² + y² = 100. Differentiate: 2x(dx/dt) + 2y(dy/dt) = 0. At x=6: y=8. So 2(6)(2) + 2(8)(dy/dt) = 0 → dy/dt = −1.5 ft/s.
Differentiate both sides of f(x,y) = C with respect to t using chain rule
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