Short answer
Related rates problems connect variables through a constraint equation. Differentiate that equation with respect to time t, substitute known values at an instant, and solve for the requested rate.
- Start with a geometric or physical relationship before differentiating.
- Differentiate with respect to time, not with respect to x.
- Substitute numbers after the time-derivative equation is formed.
Build the model from a constraint
For a sphere, volume and radius satisfy V = (4/3)pi r^3. If one changes over time, both are linked by this equation.
Draw a diagram and assign variable names with units to avoid mixing radius, height, and distance terms.
Differentiate and evaluate at one instant
Differentiate each term with respect to t using chain rule where needed. This creates dv/dt, dr/dt, or similar rates in one equation.
Then plug in the snapshot values and isolate the unknown rate.
- Write units on known rates and final answer.
- Check sign: increasing versus decreasing quantities.
- Use exact pi forms first, decimal at the end.
Expanding sphere radius rate
A balloon's volume increases at 24pi cm^3/s. Find dr/dt when r = 2 cm.
- Start with V = (4/3)pi r^3 and differentiate: dV/dt = 4pi r^2 * dr/dt.
- Substitute dV/dt = 24pi and r = 2: 24pi = 4pi*(4)*dr/dt.
- Solve: 24pi = 16pi*dr/dt, so dr/dt = 3/2 cm/s.
Result: The radius is increasing at 1.5 cm/s when r = 2 cm.
Common mistakes
- Substituting values before differentiating the relationship.
- Using the wrong geometric formula.
- Forgetting chain rule factors in time derivatives.
- Ignoring units in final rate statements.
Try one
If A = pi r^2 and dr/dt is known, what rule gives dA/dt?
Differentiate with respect to t: dA/dt = 2pi r*(dr/dt).
Sources
- OpenStax Calculus Volume 1Open textbook covering limits, derivatives, applications, and introductory integration.
- College Board AP CalculusOfficial AP Calculus AB course overview and unit structure.