O-Level Differentiation and integration: worked solution

Question

Baking powder falls onto a surface at 2π cm³ s⁻¹, forming a right circular cone whose radius is always one eighteenth of its height. Determine how fast the radius is changing 3 seconds into the pour. [Cone volume = (1/3)πr²h]

Given r = h/18, so h = 18r. V = (1/3)πr² h = (1/3)πr²(18r) = 6πr³. dV/dt = 18πr² (dr/dt), and dV/dt = 2π (constant inflow). Volume after 3 s: V = 2π × 3 = 6π cm³. Then 6πr³ = 6π → r³ = 1 → r = 1 cm. Substitute: 2π = 18π(1)² (dr/dt) → dr/dt = 2π/(18π) = 1/9 cm/s ≈ 0.111 cm/s.