A specialized cooling tank for industrial lasers is being designed in the shape of an open-topped cuboid. The tank is constructed from high-grade sheet metal. The base of the tank has a length of 3x3x3x metres and a width of xxx metres. the vertical height of the tank is hhh metres.
Given that the volume of the tank must be exactly 162 m3162\text{ m}^3162 m3:
Show that the total surface area A m2A\text{ m}^2A m2 of the metal required is given by
A=3x2+432x A = 3x^2 + \frac{432}{x} A=3x2+x432Use calculus to find the value of xxx for which AAA is stationary, giving your answer to 3 significant figures.
Calculate the value of d2Adx2\frac{d^2A}{dx^2}dx2d2A at the stationary point and hence determine whether this value of xxx minimizes the amount of metal used.
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.
