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.
425 exam-style questions on OCR A Level Maths 1.7 Differentiation, covering 1.7.1 Derivative as gradient of the tangent, 1.7.2 Gradient of the tangent at a point, 1.7.3 Sketching the gradient function, 1.7.4 Second derivatives, 1.7.5 Second derivative as rate of change of gradient, 1.7.6 Convex, concave and points of inflection (A-level only), 1.7.7 Differentiation from first principles for powers of x, 1.7.8 Differentiation from first principles for sin x and cos x (A-level only), 1.7.9 Differentiating x^n, 1.7.10 Differentiating e^(kx) and a^(kx) (A-level only), 1.7.11 Differentiating trigonometric functions (A-level only), 1.7.12 Derivative of ln x (A-level only), 1.7.13 Tangents and normals, 1.7.14 Stationary points, 1.7.15 Increasing and decreasing functions, 1.7.16 Points of inflection (A-level only), 1.7.17 Product and quotient rules (A-level only), 1.7.18 Chain rule (A-level only), 1.7.19 Parametric and implicit differentiation (A-level only), 1.7.20 Constructing differential equations (A-level only), and 1.7 Differentiation. Each one has a worked solution and a mark scheme showing where the marks go.