A specialized cooling tank for industrial lasers is being designed in the shape of an open-topped cuboid. The tank is constructed from high-grade sheet metal. The base of the tank has a length of 3x3x3x metres and a width of xxx metres. the vertical height of the tank is hhh metres.
Given that the volume of the tank must be exactly 162 m3162\text{ m}^3162 m3:
Show that the total surface area A m2A\text{ m}^2A m2 of the metal required is given by
A=3x2+432x A = 3x^2 + \frac{432}{x} A=3x2+x432Use calculus to find the value of xxx for which AAA is stationary, giving your answer to 3 significant figures.
Calculate the value of d2Adx2\frac{d^2A}{dx^2}dx2d2A at the stationary point and hence determine whether this value of xxx minimizes the amount of metal used.
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.