The concentration of a specific catalyst in a bioreactor, C C\,C mg/L, is monitored over a 12-hour production cycle. The concentration at time t t\,t hours, for 0≤t≤120 \le t \le 120≤t≤12, is modeled by the function:
C=t20(24+10t−t2)+5 C = \frac{\sqrt{t}}{20}(24 + 10t - t^2) + 5 C=20t(24+10t−t2)+5Given that C C\,C has a stationary value at t=αt = \alphat=α:
Use calculus to show that α \alpha\,α satisfies the equation
5α2−30α−24=0 5\alpha^2 - 30\alpha - 24 = 0 5α2−30α−24=0Hence find the value of α\alphaα, giving your answer to 3 decimal places.
Use further calculus to prove that C C\,C is a maximum at this value of α\alphaα.
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.
