An industrial storage hopper, open at the top, is to be fashioned from sheet metal in the shape of a cuboid. The base of the hopper is rectangular with a length twice its width. Let the width of the base be xxx metres and the height of the hopper be hhh metres. The design requires the hopper to have a fixed volume of 120 m3120\text{ m}^3120 m3.
Show that the total surface area, S m2S\text{ m}^2S m2, of the sheet metal used to construct the hopper is given by
S=2x2+360x S = 2x^2 + \frac{360}{x} S=2x2+x360Use calculus to find the value of xxx for which SSS has a stationary value, giving your answer to 3 significant figures.
Find d2Sdx2\frac{d^2S}{dx^2}dx2d2S and use this to justify that the value of xxx found in part (b) results in a minimum value for the required surface area.
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.
