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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.

(a) The geometric mean of 4 and 9 is √4·9 = √36 = 6 and their arithmetic mean is

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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

8·2 = 5

4= 1.25 ounces of wine.

2. Given a positive number b, under what conditions does there exist a rectangular carpet with

perimeter 2band area b

2?Show your work and simplify your ﬁnal answer. (3 marks)

Solution:

Denote the dimensions of the carpet by xand y. The given information implies that

xy =b

2and 2x+ 2y= 2b .

The second equation implies that y=b−xand then the ﬁrst equation becomes

x(b−x) = b

2⇒x2−b·x+b

2= 0 .

This quadratic equation has a real solution if and only if (−b)2−4·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.)

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