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Paula Maurutto

MAT102F - Introduction to Mathematical Proofs - Fall 2012 - UTM Quiz 1 (Version A) - September 21, 2012 - Solutions 1. Complete the following sentences: (2 marks) No explanation is need for this question. p p (a) The geometric mean of 4 and 9 is 4▯9 = 36 = 6 and their arithmetic mean is 1 2▯(4 + 9) = 6:5 . (b) A glass contains 5 ounces of wine and 3 ounces of water (mixed uniformly). If we remove 2 ounces of liquid from the glass and pour it into an empty cup, then the cup will contain 5 5 8▯2 = 4 = 1:25 ounces of wine. 2. Given a positive number b , under what conditions does there exist a rectangular carpet with b perimeter 2b and area 2 ? Show your work and simplify your ▯nal answer. (3 marks) Solution: Denote the dimensions of the carpet by x and y. The given information implies that b xy = and 2x + 2y = 2b : 2 The second equation implies that y = b ▯ x and then the ▯rst equation becomes x(b ▯ x) = b ) x ▯ b ▯ x +b = 0 : 2 2 This quadratic equation has a real solution if and only if (▯b) ▯ 2 ▯ 0, or equivalently: b ▯ 2, and that is the required condition. (Note: From the quadratic formula we can conclude that if b ▯ 2, then the solution(s) will be positive, and thus no additional conditions are needed.) 2 ▯3 ▯ 3. If 3
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