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 9.10 Rates of Change with exam-style questions for A Level Maths. 24 questions, 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.