A six-sided box, in the shape of a cuboid, is made from a sheet metal. The base of the box is x x\,x cm by y y\,y cm and the height of the box is x x\,x cm.
The volume of the box is 1000 cm3.
Show that the area of sheet metal, A cm2A \text{ cm}^2A cm2, is given by A=4000x+2x2\displaystyle A = \frac{4000}{x} + 2x^2A=x4000+2x2
Use calculus to find the value of x x\,x for which A A\,A is stationary.
Prove that this value of x x\,x gives a minimum value of AAA.
Calculate the minimum area of sheet metal needed to make the box.
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.