# HW 5 SOL'NS - Practice problems-solutions 1 I want you to...

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Practice problems-solutions 1) I want you to rewrite Archimedes proof for the area of the following sphere using the following facts Lemma 1: (zigzag) Let n be even and P n D ( r ) be inscribed in the disc D ( r ) with radius r . Let P rot n the inscribed body of revolution. Then | ∂P rot n | = π | AC || AB | , where | AC | is the diameter, AB is the ﬁrst part of the inscribed polygon. Lemma 2: (zigzag) Let n be even and D ( r ) Q n D ( r n ) be circumscribed polygon and r n be the new radius. Let Q rot n the body of revolution. Then | ∂Q rot n | = π (2 r n ) 2 | AB | | AC | . Lemma 3: | ∂Q rot n | - | ∂Sp | ≤ | ∂Q rot n | - | ∂P rot n | = 4 π ( r 2 n - r 2 ). Lemma 4: For every ε > 0 there exists a n 0 such that 4 a ( D ( r n )) - 4( a ( D ( r )) < ε . 1. Use the four facts and nail the reductio ad absurdum argument | ∂Sp | = 4 πr 2 . 2. Prove Lemma 3 and Lemma 4. Solution: Let us ﬁrst assume that | ∂Sp | < 4 πr 2 . According to Lemma 3 and Lemma 4, we can then ﬁnd n 0 such that | ∂Q rot n | < 4 πr 2 . Since the

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## This note was uploaded on 10/20/2011 for the course LER 110 taught by Professor Ashby during the Spring '11 term at University of Illinois, Urbana Champaign.

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HW 5 SOL'NS - Practice problems-solutions 1 I want you to...

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