Connected (related) rates of change

Method: (1) draw and label the variables as functions of time; (2) write a relation linking them; (3) differentiate both sides with respect to t; (4) substitute the values at the instant asked for; (5) solve and give the sign and units.

Worked examples

1. Expanding circle: A = πr² ⟹ dA/dt = 2πr·dr/dt. When dr/dt = 3 cm/s and r = 10 cm: dA/dt = 60π ≈ 188.5 cm²/s.
2. Ladder sliding: a 5 m ladder leans against a wall; the foot moves away at 1 m/s. x² + y² = 25 ⟹ x·dx/dt + y·dy/dt = 0. At x = 3, y = 4: 3(1) + 4(dy/dt) = 0 ⟹ dy/dt = −0.75 m/s (the top descends).
3. Filling cone: water in a conical tank with r = h/2, V = (1/3)πr²h = (π/12)h³ ⟹ dV/dt = (π/4)h²·dh/dt. With dV/dt = 2 m³/min at h = 3: dh/dt = 8/(9π) ≈ 0.283 m/min.
4. Two ships: one sails north at 20 km/h from a port, the other east at 15 km/h. After 2 h, x = 30, y = 40, s = 50 and s·ds/dt = x·dx/dt + y·dy/dt ⟹ 50·ds/dt = 30(15) + 40(20) = 1250 ⟹ ds/dt = 25 km/h.
5. Shadow / angle: a plane flies horizontally at 600 km/h at 5 km altitude; when its distance over the ground is 13 km the angle θ of elevation satisfies tan θ = 5/x ⟹ sec²θ·dθ/dt = −5x⁻²·dx/dt; with sec²θ = (13/5)², dθ/dt = −0.05 rad/min ≈ −2.87°/min.