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Question 5 A-constructibility - 10 marks This MSOS question is designed to test your under ...
Question 5 A-constructibility - 10 marks This MSOS question is designed to test your understanding of (im)possibility of construction with the help of ruler and compasses. It builds on the material of Chapter 19. Duplication of a cube and quadrature of a circle are two classical problems of Greek geometry. Let us consider their analogues, namely the problems of 'triplicating a cube' and 'cubing a ball'. Triplicating a cube means constructing a cube of three times the volume of a given cube, and cubing a ball means constructing a cube of the same volume as a given ball; in both cases, 'constructing' refers to using ruler and compasses only. We may also assume that the given cube has unit side length and the given ball has unit radius. (a) Show that a unit cube cannot be triplicated by a construction using ruler and compasses only, following the steps below. (i) Show that if a unit cube can be triplicated using ruler and compasses only, then \( u=\sqrt[3]{3} \) is a constructible number. (ii) Find the minimal polynomial for \( u \) over \( \mathbb{Q} \), and apply Corollary \( 2.8 \) of Chapter 19. (b) Show that a unit ball cannot be cubed by a construction using ruler and compasses only, following the steps below. (i) Show that if a unit ball can be cubed using ruler and compasses only, then \( v=\sqrt[3]{4 \pi / 3} \) is a constructible number. (ii) Show that \( v \) is transcendental over \( \mathbb{Q} \), and apply Theorem \( 2.7 \) of Chapter 19.