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On a rectangular coordinate plane, a circle centered at (0, 0) is insc

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Bunuel
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On a rectangular coordinate plane, a circle centered at (0, 0) is insc [#permalink] New post   21 Oct 2020, 00:15
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67% (02:52) correct 33% (03:11) wrong based on 96 sessions
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On a rectangular coordinate plane, a circle centered at (0, 0) is inscribed within a square with adjacent vertices at (\(0, −2 \sqrt{2}\)) and (\(2 \sqrt{2}, 0\)). What is the area of the region, rounded to the nearest tenth, that is inside the square but outside the circle?

A. 3

B. 3.2

C. 3.4

D. 3.6

E. 3.8





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Re: On a rectangular coordinate plane, a circle centered at (0, 0) is insc [#permalink] New post   21 Oct 2020, 02:04
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Another brillient question by Bunuel

diagonal of square is = 2√2+2√2= 4√2

Hence side of the sq will be 4 because diag in square is a√2

area of sq = 4 * 4 = 16

just pay attention to circle which is in the square , the diameter of the the circle will be side of sq. i.e = 4

Hence radius will be = 2

area of sq= pi * 2^2 = 3.14 * 4 = 12.56 = 12.6 ( nearest to tenth)

Hence desired area that is inside the square but outside the circle is = area of sq - an area of circle = 16- 12.6 = 3.4

Hence IMO answer will be C = 3.4

Please see the attachment for the diagram for clear understanding.

Bunuel wrote:
On a rectangular coordinate plane, a circle centered at (0, 0) is inscribed within a square with adjacent vertices at (\(0, −2 \sqrt{2}\)) and (\(2 \sqrt{2}, 0\)). What is the area of the region, rounded to the nearest tenth, that is inside the square but outside the circle?

A. 3

B. 3.2

C. 3.4

D. 3.6

E. 3.8





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Re: On a rectangular coordinate plane, a circle centered at (0, 0) is insc [#permalink] New post   21 Oct 2020, 02:11
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Given

    • On a rectangular coordinate plane, a circle centered at (0, 0) is inscribed within a square with adjacent vertices at (0, 2√2) and (2√2, 0).


To Find

• The area of the region, rounded to the nearest tenth, that is inside the square but outside the circle.

    Approach and Working Out


      • If you consider the diagram below,
        o The side of the square is the distance between (0, 2√2) and (2√2, 0).
        o We can calculate it as 4.
        o PQ will be 4 and thus radius will be 2.

      • Area of the required region = Area of the square – Area of the circle.
        o Area of the square = 16 square units.
        o Area of the circle = π\(2^2\) = 4π = 4×3.14 = 12.56
        o Answer = 16 – 12.56 = 3.44
        o ~ 3.4


    Correct Answer: Option C
    Attachments

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    exc4libur
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    Re: On a rectangular coordinate plane, a circle centered at (0, 0) is insc [#permalink] New post   21 Oct 2020, 05:40
    Bunuel wrote:
    On a rectangular coordinate plane, a circle centered at (0, 0) is inscribed within a square with adjacent vertices at (\(0, −2 \sqrt{2}\)) and (\(2 \sqrt{2}, 0\)). What is the area of the region, rounded to the nearest tenth, that is inside the square but outside the circle?

    A. 3
    B. 3.2
    C. 3.4
    D. 3.6
    E. 3.8


    len_sqr: sqrt[(2V2)^2+(2V2)^2]=[8+8]=4
    diameter=4, rad=2
    area sqr - circle = 4^2 - 2^2*pi,
    ... 16-4*3.14=16-12-.4-.16=4-.56=3.44
    rounded=3.4

    Ans (D)
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    Re: On a rectangular coordinate plane, a circle centered at (0, 0) is insc [#permalink] New post   21 Oct 2020, 09:37
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    Most important step is to start with a figure.

    Using the formula for diagonal of a square, we get a√2
    diagonal of square is = 2√2+2√2= 4√2, hence the side of the square is 4*4 = 16.

    Now, the diameter of the circle is nothing but the side of square.
    Hence the radius = 2
    Area of the circle = 3.14*4

    Required answer = 16-(3.14*4) = 3.44 = 3.4
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    Re: On a rectangular coordinate plane, a circle centered at (0, 0) is insc [#permalink] New post   21 Oct 2020, 10:49
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    Required:
    Area inside the square but outside the circle

    Required Area = Area of square - Area of Circle

    Area of Square:
    The verticies of square are (0,−2√2) and (2√2,0).
    From this we get length of the diagonal is 4√2 and side of square is 4
    Area of square is 16

    Area of circle:
    We know that diameter of the circle is equal to side of square.
    So diameter = 4, Radius = 2

    Area of circle = 4π , which then translates to 4*3.14 = 12.56

    Area of square - Area of circle = 16-12.56 = 3.44

    Ans: C
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    Re: On a rectangular coordinate plane, a circle centered at (0, 0) is insc [#permalink] New post   21 Oct 2020, 11:02
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    Side of Square turns out to be 4

    So Diameter of circle is equal to Side of Square = 4

    Radius of Circle = 4/2 = 2

    Desired Area = Area of Square - Area of Circle

    = 4*4 - 3.14 * 2 * 2

    = 16- 12.56 = 3.44

    Hence C
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    Re: On a rectangular coordinate plane, a circle centered at (0, 0) is insc [#permalink] New post   21 Oct 2020, 11:43
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    Side of the square = \sqrt{4*2 + 4*2} = 4


    area of square= 16
    Now, the diameter of the circle is nothing but the side of square.
    Hence the radius = 2
    Area of the circle = 3.14*4

    portion = 16-(3.14*4) = 3.4
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    Re: On a rectangular coordinate plane, a circle centered at (0, 0) is insc [#permalink] New post   21 Oct 2020, 20:03
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    Length of the square = distance between (0.-2sqrt2) and (2sqrt2,0) = 4

    Distance from the centre of the circle (0,0) to the midpoint of (0.-2sqrt2) and (2sqrt2,0) = circle radius = 2

    Area of square - Area of circle = (4*4) - (3.14*2*2) = 4*0.86 = 3.44

    IMO C
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    Re: On a rectangular coordinate plane, a circle centered at (0, 0) is insc [#permalink] New post   27 Oct 2023, 00:15
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    Re: On a rectangular coordinate plane, a circle centered at (0, 0) is insc [#permalink]
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