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.
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.