A decorative paperweight is designed as a right square-based pyramid inscribed within a solid crystal sphere of radius KKK. The vertex of the pyramid and the center of its square base both lie on the same diameter of the sphere. The pyramid has a vertical height hhh and the side length of its square base is sss, such that 0<h<2K0 < h < 2K0<h<2K.
Show that the volume, VVV, of the pyramid is given by
V=23(2Kh2−h3) V = \frac{2}{3}(2Kh^2 - h^3) V=32(2Kh2−h3)Find the maximum possible volume of the pyramid in terms of KKK. Fully justify your answer.
Practise Edexcel A Level Maths Differentiation with exam-style questions for A Level Maths. 327 questions covering 12.1 Gradients of Curves, 12.2 Differentiation from first principles, 12.3 Differentiating x^n, 12.4 Differentiating Quadratics, 12.5 Differentiating functions with two or more terms, 12.6 Gradients, Tangents and Normals, 12.7 Increasing and Decreasing Functions, 12.8 Second Order Derivatives, 12.9 Stationary Points, 12.10 Sketching Gradient Functions, and 12.11 Modelling with Differentiation, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
