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.
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.