An industrial chemical reactor vessel with a circular cross-section and a maximum depth of 30 cm is initially empty. A cooling reagent is pumped into the reactor such that its depth at time ttt seconds is hhh cm.
The volume of reagent in the reactor, V cm3V \text{ cm}^3V cm3, is modelled by the formula:
V=112h2(2h+45)0≤h≤30 V = \frac{1}{12}h^2(2h + 45) \quad 0 \le h \le 30 V=121h2(2h+45)0≤h≤30The reagent is delivered at a constant rate of 225 cm3 s−1225 \text{ cm}^3\text{ s}^{-1}225 cm3 s−1. According to this model:
Determine the time required to fill the reactor vessel completely.
Calculate the rate of change of the depth of the reagent, in cm s−1\text{cm s}^{-1}cm s−1, at the instant the depth reaches 15 cm.
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.
