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 AQA A Level Maths 1.10 G: Differentiation with exam-style questions for A Level Maths. 321 questions 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), matched to the AQA A Level Maths (7357) 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.
