An aquaculture company is designing a rectangular treatment tank with an open top. The tank is to have a base where the length is exactly twice the width. Let the width of the base be www metres and the height of the tank be hhh metres.
Given that the capacity of the tank is fixed at 288 m3288 \text{ m}^3288 m3:
Show that the total internal surface area, S m2S \text{ m}^2S m2, of the tank is given by
S=2w2+864w S = 2w^2 + \frac{864}{w} S=2w2+w864Use algebraic differentiation to find the value of www for which SSS has a stationary point.
Justify by further differentiation that this value of www gives a minimum internal surface area for the tank.
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.