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Differentiation

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Question 79

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)+5

Given that C C\,C has a stationary value at t=αt = \alphat=α:

a.

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=0
[4]
b.

Hence find the value of α\alphaα, giving your answer to 3 decimal places.

[2]
c.

Use further calculus to prove that C C\,C is a maximum at this value of α\alphaα.

[3]
Markscheme

Differentiation Questions

  1. A Level
  2. /Maths
  3. /Differentiation

717 exam-style questions on Edexcel A Level Maths Differentiation, covering 9.1 Differentiating sin x and cos x, 9.2 Differentiating exponentials and logarithms, 9.3 The Chain Rule, 9.4 The Product Rule, 9.5 The Quotient Rule, 9.6 Differentiating Trigonometric Functions, 9.7 Parametric Differentiation, 9.8 Implicit Differentiation, 9.9 Using Second Derivatives, and 9.10 Rates of Change. Each one has a worked solution and a mark scheme showing where the marks go.

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