Linked changes
In related rates problems, several quantities change over time, and an equation connects them. Knowing one rate of change lets you find another.
Example: As a balloon inflates, its radius and volume both change. If you know how fast air goes in (dV/dt), how fast does the radius grow (dr/dt)?
The process
- Draw a diagram; label variables (things that change) and constants.
- Write an equation linking the variables.
- Differentiate both sides with respect to time t (implicit differentiation, using the chain rule).
- Substitute values at the given instant.
- Solve for the unknown rate.
Important: Substitute specific values only after differentiating.
Example 1: inflating balloon
Air is pumped in at dV/dt = 100 cm³/s. How fast is the radius growing when r = 5 cm?
- V = (4/3)πr³
- dV/dt = 4πr² dr/dt
- 100 = 4π(25) dr/dt
- dr/dt = 1/π ≈ 0.32 cm/s
Example 2: sliding ladder
A 10 m ladder slides down a wall. The bottom moves away at 1 m/s. How fast is the top falling when the bottom is 6 m from the wall?
- x² + y² = 100
- 2x dx/dt + 2y dy/dt = 0
- When x = 6: y = 8
- 2(6)(1) + 2(8) dy/dt = 0 → dy/dt = −0.75 m/s
The top falls at 0.75 m/s.
Example 3: draining cone
Water drains from a cone (point down) with radius 4 m and height 8 m. When the water depth is 6 m, the level falls at 0.5 m/min. How fast is the volume changing?
- Similar triangles: r = h/2
- V = (1/3)πr²h = (π/12)h³
- dV/dt = (π/4)h² dh/dt
- = (π/4)(36)(−0.5) = −4.5π ≈ −14.1 m³/min
Example 4: spreading circle
A ripple’s radius grows at 2 cm/s. How fast is the area growing when r = 10 cm?
- A = πr² → dA/dt = 2πr dr/dt = 2π(10)(2) = 40π ≈ 126 cm²/s
Signs matter
- Increasing quantities have positive rates.
- Decreasing quantities (draining, falling) have negative rates.
Worked examples
Example 1: A square’s side grows at 3 cm/s. dA/dt when s = 5? 2(5)(3) = 30 cm²/s.
Example 2: A cube’s edge grows at 0.1 cm/s. dV/dt when s = 4? 3(16)(0.1) = 4.8 cm³/s.
Example 3: Two cars leave an intersection: one north at 60 km/h, one east at 80 km/h. How fast is the distance between them growing? (Constant here: 100 km/h.)
Common mistakes
- Substituting values before differentiating.
- Forgetting the chain rule (each variable gets a d/dt).
- Sign errors for decreasing quantities.
Try it yourself
- A circle’s radius grows at 3 m/s. dA/dt when r = 4?
- A sphere’s volume grows at 50 cm³/s. dr/dt when r = 10?
- A 13 m ladder’s bottom slides out at 2 m/s. How fast does the top fall when the bottom is 5 m out?
Answers: 1) 24π ≈ 75.4 m²/s 2) 1/(8π) ≈ 0.040 cm/s 3) 5/6 ≈ 0.83 m/s
Quick recap
- Related rates link changing quantities with an equation.
- Differentiate with respect to t (implicit differentiation), then substitute.
- Use correct signs and units for each rate of change.