A precision-engineered storage vault for high-density alloys is in the shape of a cuboid with a rectangular base of width xxx cm and length 3x3x3x cm. The height of the vault is hhh cm.
The volume of the vault is fixed at 1350 cm31350\text{ cm}^31350 cm3.
Show that the surface area of the vault, S cm2S\text{ cm}^2S cm2, is given by
S=6x2+3600x S = 6x^2 + \frac{3600}{x} S=6x2+x3600Find dSdx\frac{dS}{dx}dxdS.
Hence find the value of xxx for which SSS is stationary, giving your answer to 3 significant figures.
Find d2Sdx2\frac{d^2S}{dx^2}dx2d2S and hence show that the value of xxx found in part (c) gives the minimum value of SSS.
Hence find the minimum surface area of the vault, giving your answer to 1 decimal place.
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.