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Related Rates

Related Rates

A class of problems in which rates of change are related by means of differentiation. Standard examples include water dripping from a cone-shaped tank and a man’s shadow lengthening as he walks away from a street lamp.

 

Example: Leaky inverted cone tank, height 4m, base radius 2m, leak rate 0.15 m³/sec. Find dh/dt when h=0.7m.
Inverted cone tank with height h, radius r=h/2. Solution derives V=(1/12)πh³, then dh/dt≈-0.39 m/sec when h=0.7.

Worked Example

Problem: A circular oil spill is expanding so that its radius increases at a constant rate of 2 meters per second. How fast is the area of the spill increasing when the radius is 5 meters?
Step 1: Write the equation relating area and radius of a circle.
A=πr2A = \pi r^2
Step 2: Differentiate both sides with respect to time tt using the chain rule.
dAdt=2πr⋅drdt\frac{dA}{dt} = 2\pi r \cdot \frac{dr}{dt}
Step 3: Substitute the known values: r=5r = 5 m and drdt=2\frac{dr}{dt} = 2 m/s.
dAdt=2π(5)(2)=20π\frac{dA}{dt} = 2\pi(5)(2) = 20\pi
Answer: The area is increasing at a rate of 20π≈62.820\pi \approx 62.8 square meters per second when the radius is 5 meters.

Why It Matters

Related rates problems model real situations where multiple quantities change simultaneously — for example, how fast a balloon's volume grows given how fast its radius increases, or how quickly a ladder's top slides down a wall as its base moves outward. They are among the most common applications of implicit differentiation in a first calculus course.

Common Mistakes

Mistake: Substituting known values before differentiating.
Correction: If you plug in a specific number for a changing quantity before you differentiate, you turn that variable into a constant and lose its rate of change. Always differentiate the general equation first, then substitute.

Related Terms