The depth of water in a reservoir, DDD metres, was recorded over a 10-day period. The depth at time ttt days, where 0≤t≤100 \le t \le 100≤t≤10, is modeled by the equation:
D=t30(18+8t−t2)+12 D = \frac{\sqrt{t}}{30}(18 + 8t - t^2) + 12 D=30t(18+8t−t2)+12Given that DDD has a stationary value at t=αt = \alphat=α:
Use calculus to show that α\alphaα satisfies the equation
5α2−24α−18=0 5\alpha^2 - 24\alpha - 18 = 0 5α2−24α−18=0Hence find the value of α\alphaα, giving your answer to 3 decimal places.
Use further calculus to prove that DDD is a maximum at this value of α\alphaα.
Practise Edexcel A Level Maths Differentiation with exam-style questions for A Level Maths. 311 questions 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, 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.
