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.
717 exam-style questions on Edexcel A Level Maths Differentiation, 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. Each one has a worked solution and a mark scheme showing where the marks go.