A large industrial hopper in the shape of an inverted square-based pyramid is being filled with fine grain at a constant rate of 24 cm3/s24\text{ cm}^3/\text{s}24 cm3/s.
After ttt seconds, the depth of the grain in the hopper is h cmh\text{ cm}h cm. The apex of the pyramid is at the bottom.
When the depth of the grain is h cmh\text{ cm}h cm, the volume V cm3V\text{ cm}^3V cm3 of the grain is given by
V=29h3 V = \frac{2}{9}h^3 V=92h3Show that when t=3t = 3t=3,
dVdh=12183 \frac{dV}{dh} = 12\sqrt[3]{18} dhdV=12318Hence, find the rate at which the depth of the grain is increasing when t=3t = 3t=3. Give your answer to three significant figures.
Practise Edexcel A Level Maths Differentiation with exam-style questions for A Level Maths. 327 questions covering 12.1 Gradients of Curves, 12.2 Differentiation from first principles, 12.3 Differentiating x^n, 12.4 Differentiating Quadratics, 12.5 Differentiating functions with two or more terms, 12.6 Gradients, Tangents and Normals, 12.7 Increasing and Decreasing Functions, 12.8 Second Order Derivatives, 12.9 Stationary Points, 12.10 Sketching Gradient Functions, and 12.11 Modelling with Differentiation, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
