A grain hopper dispenses wheat at a constant rate of 12 cm3s-1 onto a level floor, where it forms a conical pile. After t t\,t seconds, the pile has a height of h cmh\text{ cm}h cm and a volume of V cm3V\text{ cm}^3V cm3. The volume is modeled by the equation:
V=πh312 V = \frac{\pi h^3}{12} V=12πh3Show that when t=6t = 6t=6,
dVdh=916π3 \frac{dV}{dh} = 9 \sqrt[3]{16\pi} dhdV=9316πHence, find the rate at which the height of the pile is increasing when t=6t = 6t=6. Give your answer in cm s−1\text{cm s}^{-1}cm s−1 to three significant figures.
375 exam-style questions on AQA A Level Maths 1.10 G: Differentiation, covering 1.10.1 The derivative and second derivative, 1.10.2 Differentiating standard functions, 1.10.3 Applications of differentiation, 1.10.4 Product, quotient and chain rules (A-level only), 1.10.5 Implicit and parametric differentiation (A-level only), and 1.10.6 Constructing differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.