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.