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Two strings have equal length. One of the strings is shaped, with no
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Bunuel
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Two strings have equal length. One of the strings is shaped, with no [#permalink] New post  22 May 2020, 10:17
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Two strings have equal length. One of the strings is shaped, with no overlap, into a circle. The other string is shaped, with no overlap, into a square. What is the ratio of the area of the region enclosed by the circle to the area of the region enclosed by the square?


A. \(1:2\)

B. \(\pi:2\)

C. \(1:1\)

D. \(4:\pi\)

E. The ratio depends on the length of the two strings.


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nick1816
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Two strings have equal length. One of the strings is shaped, with no [#permalink] New post  22 May 2020, 11:16
Suppose the length of the strings = \(4\pi\)

Perimeter of the square = \(4\pi\)
Length of the side of square = \(\pi\)
Area of square= \({\pi}^2\)

Perimeter of the square =\( 4\pi\)
Radius of the side of square = [\(\frac{4\pi}{2\pi} =2\)
Area of square= \({\pi}*2^2\) = \(4\pi\)

ratio of the area of the region enclosed by the circle to the area of the region enclosed by the square = \(4\pi\) : \({\pi}^2\) = \(4:\pi\)

Bunuel wrote:
Two strings have equal length. One of the strings is shaped, with no overlap, into a circle. The other string is shaped, with no overlap, into a square. What is the ratio of the area of the region enclosed by the circle to the area of the region enclosed by the square?


A. \(1:2\)

B. \(\pi:2\)

C. \(1:1\)

D. \(4:\pi\)

E. The ratio depends on the length of the two strings.


PS20405
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Enchanting
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Re: Two strings have equal length. One of the strings is shaped, with no [#permalink] New post  22 May 2020, 11:28
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Bunuel wrote:
Two strings have equal length. One of the strings is shaped, with no overlap, into a circle. The other string is shaped, with no overlap, into a square. What is the ratio of the area of the region enclosed by the circle to the area of the region enclosed by the square?


A. \(1:2\)

B. \(\pi:2\)

C. \(1:1\)

D. \(4:\pi\)

E. The ratio depends on the length of the two strings.
PS20405

Solution:
    Length of the two strings are equal, say l.
      Circumference of circle = l
      2*π*r=l, where r is the radius of circle.
      \(r=\frac{l}{2π}\)
        Area= \(π*r^2=π*\frac{l^2}{(4π^2 )}=\frac{l^2}{4π}\)
    Perimeter of square = l
    Side of the square = \(\frac{l}{4}\)
      Area = \((\frac{l}{4})^2=\frac{l^2}{16}\)
    Required ratio = \(\frac{\frac{l^2}{4π}}{\frac{l^2}{16}}=\frac{4}{π}\)
Answer: Option D.
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Re: Two strings have equal length. One of the strings is shaped, with no [#permalink] New post  22 May 2020, 18:53
let strings be 12 each
so for circle ; 2 pi r= 12 ; r = 6/pi
and for square ; 4s = 12 ;s =3
area of circle ; 36/pi
and area of square = 9
ratio of the area of the region enclosed by the circle to the area of the region enclosed by the square
36/pi * 1/9 = 4:pi
OPTION D

Bunuel wrote:
Two strings have equal length. One of the strings is shaped, with no overlap, into a circle. The other string is shaped, with no overlap, into a square. What is the ratio of the area of the region enclosed by the circle to the area of the region enclosed by the square?


A. \(1:2\)

B. \(\pi:2\)

C. \(1:1\)

D. \(4:\pi\)

E. The ratio depends on the length of the two strings.


PS20405
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NitishJain
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Posts: 296
Re: Two strings have equal length. One of the strings is shaped, with no [#permalink] New post  23 May 2020, 16:57
D?

Let the total length = l
Side of square = a
radius of circle = r

length of the string should be equal to the perimeter of the square
l = 4a
or a = l/4

Similarly, length of the string should be equal to the perimeter of the circle
l = 2(pie)r
r = l/2(pie)

Area of circle : Area os Square = (pie) r^2 : a^2

Substituting values of 'r' and 'a' in terms of 'l':
Area of circle : Area os Square = 4: (pie)
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koam7
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Re: Two strings have equal length. One of the strings is shaped, with no [#permalink] New post  23 May 2020, 17:49
Let the length of the string be x.

The length of each side of the square will be x/4, so the area of the square will be x^2/16

The circumference of the circle will be x, so 2(pi)R = x, or R = x / 2(pi). So, the area of the circle will be pi * x^2/4(pi^2), or x^2/4pi

The ratio of the area of the circle to the square will be

x^2/4pi : x^2/16

eliminate the x^2

1/4pi : 1/16

Simplified, this gives 4 : pi

So, the answer is D
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Basshead
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Re: Two strings have equal length. One of the strings is shaped, with no [#permalink] New post  28 Nov 2020, 11:59
Let \(x = length of each string\)

\frac{x}{4} = side of a square

\(\frac{x}{4}^2 = \frac{x^2}{16}\)

\(x = circumference\)

\(x = 2πr\)

\(\frac{x}{2π} = r\)

\(\frac{x^2}{4π^2} = r^2\)

\(\frac{x^2}{4π} = πr^2\)

\(\frac{area of circle}{area of square} = \frac{16}{4π} = \frac{4}{π}\)

Answer is D.
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Re: Two strings have equal length. One of the strings is shaped, with no [#permalink]
28 Nov 2020, 11:59
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