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+360xS = 2x^2 + \frac{360}{x}S=2x2+x360
Use 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 9.9 Using Second Derivatives with exam-style questions for A Level Maths. 57 questions, 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.