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Suppose five circles, each 4 inches in diameter, are cut from a rectan

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Suppose five circles, each 4 inches in diameter, are cut from a rectan  [#permalink]

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New post 14 Sep 2011, 12:27
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Suppose five circles, each 4 inches in diameter, are cut from a rectangular strip of paper 12 inches long. If the least amount of paper is to be wasted, what is the width of the paper strip?

(A) 5
(B) 4+2\(\sqrt{3}\)
(C) 8
(D) 4(1+\(\sqrt{3}\))
(E) 12

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Re: Suppose five circles, each 4 inches in diameter, are cut from a rectan  [#permalink]

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New post 11 Sep 2014, 22:35
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jamifahad wrote:
Suppose five circles, each 4 inches in diameter, are cut from a rectangular strip of paper 12 inches long. If the least amount of paper is to be wasted, what is the width of the paper strip?

(A) 5
(B) 4+2\(\sqrt{3}\)
(C) 8
(D) 4(1+\(\sqrt{3}\))
(E) 12


First, think what they mean by 'least amount of paper is to be wasted'? Since the dimension of the circles is given, perhaps what they mean is that the width of the paper used must be minimum since the leftover paper will be wasted. So how will you place the 5 circles? The diameter of the circles is 4 each and the length of the paper is 12 so you can place 3 circles neatly across the length. The other two must be placed in a way which reduces width of the paper, which means between two circles rather than edge to edge.

Attachment:
Ques4.jpg
Ques4.jpg [ 23.97 KiB | Viewed 2474 times ]

Once the diagram is ready, the solution is self explanatory. The equilateral triangle in the middle has side 4 because the sides are formed by 2 radii. The width of the paper is the sum of dashed lines, the regular ones are 2 each (radii of circles) and the bold one is the altitude of the equilateral triangle \(= \sqrt{3}*4/2 = 2*\sqrt{3}\).

So width \(= 2 + 2*\sqrt{3} + 2 = 4 + 2*\sqrt{3}\)
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Re: Suppose five circles, each 4 inches in diameter, are cut from a rectan  [#permalink]

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New post 14 Sep 2011, 19:49
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Well, don't know how to draw diagrams here but let me see if i can explain through text.

First Approach (Traditional): First place 3 circles back-to-back (touching each other) to make it for 12 inches length. Now for least wastage of paper, you need to put 4th and 5th circles above 3 circles. 4th between 1st and 2nd circles and 5th between 2nd and 3rd. Now you can make equilateral triangle by joining centers of 1st, 2nd, and 4th circles. So total width of the strip will be radius of 4th cricle + height of equilateral triangle + radius of 1st (or 2nd) circle. It equals 2 + (sqrt3/2)4 + 2 = 4 + 2sqrt3

Second approach (quicker): Just think, if we place 4th circle exactly on the top of the 1st circle width of the strip will be sum of diameters of 2 circles 1.e 8. But that is not the case. Since 4th circle is placed between 1st and 2nd, some part of their diameters will overlap. It means our answer should be less than 8. You can eliminate options (C), (D), and (E). Option (A) is too less a value (it is suggesting that 4th circle is actually fitting 3/4th of its diameter between 1st and 2nd circles, which is not feasible). So you are left with only option and that's your answer.

I hope it helps!!!
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Suppose five circles, each 4 inches in diameter, are cut from a rectan  [#permalink]

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New post 23 Aug 2017, 12:43
If we are cutting five circles each with radius of 2, their total area is 5*4*pi which is slightly greater than 60.
Length of Strip = 12
So breadth should be more than 5. Since least paper is wasted
Option A: Wrong
Option B: Most suitable
Option C: Bigger then option B
Option D: Bigger than option B
Option E: Bigger than option B
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Suppose five circles, each 4 inches in diameter, are cut from a rectan   [#permalink] 23 Aug 2017, 12:43
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