Related Rates
Related rates problems connect the rates of change of two or more quantities that are linked by an equation, using implicit differentiation with respect to time.
The related rates recipe
1) Draw a picture and label variables. 2) Write an equation relating the variables (often geometry: area, volume, Pythagorean theorem). 3) Differentiate both sides with respect to time t, using the chain rule for every variable. 4) Plug in the known values (only after differentiating!) and solve for the unknown rate.
The most common error is substituting numbers in before differentiating — that freezes a variable's rate at 0. Differentiate the general relationship first, then substitute specific values.
Worked Example
A ladder 10 ft long leans against a wall. The bottom slides away from the wall at 2 ft/s. How fast is the top sliding down when the bottom is 6 ft from the wall?
- Let x = distance of bottom from wall, y = height of top. Relationship: x² + y² = 100 (Pythagorean theorem, ladder length fixed at 10).
- Differentiate with respect to t: 2x(dx/dt) + 2y(dy/dt) = 0.
- When x=6: y² = 100-36=64, so y=8. Given dx/dt = 2.
- Substitute: 2(6)(2) + 2(8)(dy/dt) = 0 → 24 + 16(dy/dt) = 0 → dy/dt = -24/16 = -1.5
Answer: The top slides down at 1.5 ft/s (negative sign shows y is decreasing).
Key terms (1)
- Related rates
- Problems that use implicit differentiation with respect to time to relate the rates of change of connected quantities.
Practice Quiz
A circle's radius grows at 3 cm/s. How fast is the area growing when r = 5 cm? (A = πr²)