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Thread: The diameter of the circle...

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    The diameter of the circle...

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    The diameter of the circle is 20 and the area of the shaded region is 80 pi. What is the value of a + b + c + d

    A) 144
    B) 216
    C) 240
    D) 270
    E) 288




    How do you solve these fast?

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    wow #1 seems pretty hard, i dont know how to solve it either.

    for #2, represent the radius of circle Q with r.
    col A = area of circle O - area of white region
    col B = area of white region * 4

    so it's best to start with getting the area of white region.
    area of white region = Pi * (2r)^2 - Pi * r^2 = 3r^2 * Pi

    col A = Pi * (4r)^2 - area of white region
    = 16r^2 Pi - 3r^2 Pi
    = 13r^2 Pi

    col B = 3r^2 Pi * 4
    = 12r^2 Pi

    Therefore, col A > col B

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    Btw OA for first is 288

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    Its better that we assume values for these type of problems.
    Let OA = 4; so OP = 2; PQ = 1

    Now comparison is between

    pi(4)^2 + pi (1)^2 - pi ( 2^2 - 1^2) and 4[ pi(2^2 - 1^2]

    14pi Vs 12pi

    col A is greater

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    Trying to make mom and pop proud Sandeep Bansal's Avatar
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    Hi ALL,

    What is the solution for the first one?can some one explain?
    My GRe is on 3rd...please help

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    Quote Originally Posted by Sandeep Bansal View Post
    What is the solution for the first one?can some one explain?
    My GRe is on 3rd...please help

    Solution is 288.
    How is still a mystery

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    An Urch Guru Pundit Swami Sage
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    Arrow Question No. 1

    Hi,
    The question needs some clarifications. It should mention that two vertices of the traingles meet at the centre of the circle.
    Total area of the circle=100pi
    Area of the two traingle=100pi-80pi=20pi
    Angle at the vertex (of one traingle)= (360*20pi*1/2)/100pi
    = 36 degree
    so, a+b+c+d=(180-36)+(180-36)=2800


    Regards,

    Jakir

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    Trying to make mom and pop proud Sandeep Bansal's Avatar
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    Question Hi , I have got the solution

    Answer is 288

    There are two ways to get it
    easy by approximation as angle subtended at the center has nothing to do with the area of traingle , but with the sector....
    1)
    See it is told that area of shaded region is 80pi
    and area of circle is pi*(R^2)=pi*100
    so the area of traingle is 100pi - 80 pi= 20pie
    now the two chords are diameters if they intersect, they at the centre and bisect equally and engendering two equal trianlges....
    approx(area of one trinagle will be ) 10pi

    in circle total area is find out by sector ;;;;
    if we can fine the sector we can get the answer...
    ratio of the areas=20pi/100pi(100pi is the total area , full circle)
    that is ratio of angles should be for two triangles (or so called sector) --->360*(2/10)=72
    for 1 sector angle should be 360*(1/10)=36
    A+B+C+D+2x(say)=180+180
    A+B+C+D= 360-2*36
    A+B+C+D=288
    Hence proved without interring further into the formula.....


    Other probable solution is
    Let E be the thrid angle in the traingle

    1/2*radius*radiusSinA=area of traingle
    area of unshaded cirle comes out to be=100pie-80pie=20pie
    so,
    E=180-(c+d),180-(a+b)

    1/2*100*SinE + 1/2*100*SinE =20pie
    E comes out to be 39 near about
    so
    360 -(39+39) = 282 (approx) = 288


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    Hi, KBTA, i see that u used the angle and area relationship to solve question 1. I have a question though.

    Let's say angle E is the the third angle of the triangle with angles C and D.

    The equation should be E/360 = area of circular region covered by E / area of circle, right?

    The area of the circular region covered by E includes the small unshaded triangle, and a small shaded region. However, in your equation, you seem to have ignored the area of the shaded part, and only considered the area of the small triangle (20pi*1/2). Could you explain why?

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    Trying to make mom and pop proud Sandeep Bansal's Avatar
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    Hi yeah ,

    I have not ignored this section that is why wrote, by approximation , the angle of the sector(which we need to know) has nothing to do with the area of traingle......
    so by approximation the answer should be....
    as there is no way except ,you do lenghty calculations........
    the 2nd solution to this problem.......
    please let me know other solution for this, I am not able to descry the figure, and found it a little ambigous.....i guess we have to reach the best possible answer to get it going.....




    Quote Originally Posted by zymeth02 View Post
    Hi, KBTA, i see that u used the angle and area relationship to solve question 1. I have a question though.

    Let's say angle E is the the third angle of the triangle with angles C and D.

    The equation should be E/360 = area of circular region covered by E / area of circle, right?

    The area of the circular region covered by E includes the small unshaded triangle, and a small shaded region. However, in your equation, you seem to have ignored the area of the shaded part, and only considered the area of the small triangle (20pi*1/2). Could you explain why?

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