← Differentiation methods

8 Related rates

Two quantities change with time; link them with an equation, then differentiate with respect to t (chain rule) to relate their rates.

Method

• Write the equation relating the variables (e.g. V and r).
• Differentiate both sides with respect to t.
• Substitute the known values and the known rate; solve for the unknown rate.

Worked example Expanding balloon A sphere’s volume grows at 100 cm³ s⁻¹. How fast is the radius growing when r = 5 cm? (V = ⁴⁄₃πr³) dV/dt = 4πr² · (dr/dt) (differentiate V wrt t). 100 = 4π(5²)(dr/dt) = 100π (dr/dt). dr/dt = 100 / (100π) = 1/π. ✓ dr/dt = 1/π ≈ 0.318 cm s⁻¹. Worked example When the given rate is an angle A camera is fixed 40 m from a straight running track, filming a runner. When the runner is 30 m from the point on the track closest to the camera, the angle θ between the camera's line of sight and that closest-point direction is increasing at 0.05 rad s⁻¹. Find the runner's speed at that instant. (tan θ = distance along track ÷ 40) Let x = distance along the track from the closest point. tan θ = x/40, so x = 40 tan θ. Differentiate with respect to t: dx/dt = 40 sec²θ · (dθ/dt). When x = 30: tan θ = 30/40 = 3/4, so sec²θ = 1 + tan²θ = 1 + 9/16 = 25/16. dx/dt = 40 × (25/16) × 0.05 = 3.125. ✓ The runner is moving at 3.125 m s⁻¹.

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