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Calculus
Related Rates
How linked quantities change together over time
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Related rates problems use calculus to connect quantities that change together over time. They matter because many real situations involve linked motion, such as a ladder sliding, a balloon rising, or a shadow growing. In the ladder example, the wall, floor, and ladder form a right triangle whose side lengths change while the ladder length stays fixed. Differentiation lets us turn a geometric relationship into an equation between rates.
Key Facts
- For a sliding ladder, x(t)^2 + y(t)^2 = L^2.
- Differentiate with respect to time: 2x dx/dt + 2y dy/dt = 0.
- Simplified ladder rate equation: x dx/dt + y dy/dt = 0.
- If the ladder length L is constant, then dL/dt = 0.
- Pythagorean theorem often provides the starting equation for right-triangle related rates.
- Signs matter: if x increases then dx/dt > 0, and if y decreases then dy/dt < 0.
Vocabulary
- Related rates
- A calculus method for finding how one changing quantity's rate depends on another changing quantity's rate.
- Derivative with respect to time
- A derivative such as dx/dt that measures how fast a variable changes as time passes.
- Implicit differentiation
- A method of differentiating an equation containing related variables without first solving for one variable.
- Constant length
- A quantity such as ladder length L that does not change with time, so its rate of change is zero.
- Rate sign
- The positive or negative sign of a rate that shows whether a quantity is increasing or decreasing.
Common Mistakes to Avoid
- Forgetting to differentiate with respect to time is wrong because x and y are functions of t, so d(x^2)/dt becomes 2x dx/dt, not just 2x.
- Treating the ladder length as changing is wrong when the ladder is rigid, because L is constant and dL/dt = 0.
- Ignoring negative signs is wrong because downward motion gives dy/dt < 0 and outward motion gives dx/dt > 0.
- Substituting numbers before differentiating can be wrong because it may turn changing variables into constants and destroy the rate relationship.
Practice Questions
- 1 A 10 m ladder leans against a wall. The bottom slides away from the wall at 0.5 m/s. When the bottom is 6 m from the wall, how fast is the top sliding down?
- 2 A 13 ft ladder slides down a wall. When the top is 12 ft above the ground, the top is moving downward at 2 ft/s. How fast is the bottom moving away from the wall?
- 3 In a sliding ladder problem, explain why the top moving downward and the bottom moving outward must have opposite signs in the equation x dx/dt + y dy/dt = 0.