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.
375 exam-style questions on AQA A Level Maths 1.10 G: Differentiation, covering 1.10.1 The derivative and second derivative, 1.10.2 Differentiating standard functions, 1.10.3 Applications of differentiation, 1.10.4 Product, quotient and chain rules (A-level only), 1.10.5 Implicit and parametric differentiation (A-level only), and 1.10.6 Constructing differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.