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geometry

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priyank.hirani Just gettin' started! Default Avatar
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geometry Post Sun Apr 15, 2012 12:14 pm
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  • Lap #[LAPCOUNT] ([LAPTIME])
    How to solve
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    Neo Anderson Rising GMAT Star Default Avatar
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    Post Sun Apr 15, 2012 12:24 pm
    from your figure: AB/AC = x / R
    as it is given that AB = BC thus AC = 2 AB thus we have R = 2x

    Volume of right side of cone = 1/3*Pi*x^2*AB ----(I)
    Volume of left side of cone = volume of full cone - volume of right side of the cone.
    = 1/3*Pi*R^2*AC - 1/3*Pi*x^2*AB
    as R = 2x & AC = 2 AB
    = 1/3*Pi(2x)^2*2*AB - 1/3*Pi*x^2*AB = 1/3*Pi*x^2*AB (2*2^2 - 1) = 1/3*Pi*x^2*AB (8 - 1) = 1/3*7*Pi*x^2*AB ---(II)

    The required ratio is the Ratio of (I) and (II) = 1/7



    Last edited by Neo Anderson on Sun Apr 15, 2012 12:32 pm; edited 1 time in total

    shubham_k Just gettin' started! Default Avatar
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    Post Sun Apr 15, 2012 12:29 pm
    this is how i approached the problem...
    let r be the radius of the base of the orginal cone and h be the height...

    thus the smaller cone will have a radius of r/2 and height h/2.

    volume of the cone = 1/3 pi*r^2h

    1/3*pi (r/2)^2*h/2= pi*r^2*h/24...................................1


    the remaining part will

    1/3pi*r^2h- pi*r^2*h/24

    = 7*pi*r^2*h/24.....................................................2

    ratio of 1 by 2

    (pi*r^2*h/24)/(7*pi*r^2*h/24)

    which is 1:7.

    please let me know if u have any doubts..

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    Post Sun Apr 15, 2012 1:14 pm


    In the figure above, the values in red have been added to the original drawing.
    The question asks for the ratio of the left half of the cone to the right half of the cone.
    (Please note that, in this context, half does NOT mean 1/2 the volume.)

    The two cones are similar.
    It is given that AB=BC.
    Thus, AC = 2AB.
    Since the height of the larger cone is twice the height of the smaller cone, every dimension of the larger cone -- including the radius -- is twice the corresponding dimension of the smaller cone.

    We need to determine the ratio of the volumes.
    V = (1/3)∏r²h.
    Since each volume includes a factor of 1/3 and a factor of ∏, the ratio of the volumes is not affected by these factors
    Thus, we can determine the ratio of the volumes by considering only the two other factors in each volume: r²h.

    We can plug in values for the two radii and the two heights.

    Smaller cone (the right half):
    Let r=1 and h=1.
    r²h = (1²)(1)= 1.

    Larger cone:
    Since each dimension is twice as long, r=2 and h=2.
    r²h = (2²)(2)= 8.

    Left half of the cone:
    Larger cone - smaller cone = 8-1 = 7.

    (right half) : (left half) = 1:7.

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    Post Sun Apr 15, 2012 1:22 pm
    priyank.hirani wrote:
    How to solve
    => Volume of right half cone/Volume of left half cone =

    = Volume of right half/Volume of full cone- Volume of right half cone

    = [(S/4)*(H/2)/3]/[SH/3 - (S/4)*(H/2)/3] = 1/7.

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