A transport container for radioactive waste is designed as a cuboid with width xxx m, length 2.5x2.5x2.5x m, and height hhh m. The total interior volume of the container must be 500 m3500\text{ m}^3500 m3.
Show that the total surface area of the container, S m2S\text{ m}^2S m2, is given by
S=5x2+1400x S = 5x^2 + \frac{1400}{x} S=5x2+x1400Find dSdx\frac{\text{d}S}{\text{d}x}dxdS.
Hence find the value of xxx for which SSS is stationary, giving your answer to 3 significant figures.
Find d2Sdx2\frac{\text{d}^2S}{\text{d}x^2}dx2d2S and hence verify that the value of xxx found in part (c) gives a minimum value for SSS.
Calculate the minimum surface area of the container, 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.