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Math Help - maximize volume of cone in sphere

  1. #1
    Gul
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    maximize volume of cone in sphere

    A cone is inscribed in a sphere of radius 1, find the dimensions of the cone, and the maximum volume of the cone?

    I get a right right triangle:


    (h-1)^2 + r^2 = 1^2
    h^2 - 2h +1 + r^2 = 1

    r^2 = -h^2 + 2h

    r^2 = 2h - h^2

    Volume is: 1/3*pi*r^2*h

    V = 1/3*pi*(2h - h^2 )*h
    V = 1/3*pi*(2h^2-h^3)
    V' = 1/3*pi(4h - 3h^2) = 0
    V' = 4h - 3h^2 = 0 , h(4-3h) = 0 , h = 4/3

    r^2 = 2h - h^2

    r^2 = 2(4/3) - (4/3)^2

    r^2 = 8/3 - 16/9
    r^2 = 8/9
    r = sqrt(8/9)
    r = .94

    Therefore Vmax = 1/3*pi*(8/9)*(4/3) = 1.241 cm^3


    Have i gone wrong somwhere
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  2. #2
    Flow Master
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    Quote Originally Posted by Gul View Post
    A cone is inscribed in a sphere of radius 1, find the dimensions of the cone, and the maximum volume of the cone?

    I get a right right triangle:


    (h-1)^2 + r^2 = 1^2
    h^2 - 2h +1 + r^2 = 1

    r^2 = -h^2 + 2h

    r^2 = 2h - h^2

    Volume is: 1/3*pi*r^2*h

    V = 1/3*pi*(2h - h^2 )*h
    V = 1/3*pi*(2h^2-h^3)
    V' = 1/3*pi(4h - 3h^2) = 0
    V' = 4h - 3h^2 = 0 , h(4-3h) = 0 , h = 4/3

    r^2 = 2h - h^2

    r^2 = 2(4/3) - (4/3)^2

    r^2 = 8/3 - 16/9
    r^2 = 8/9
    r = sqrt(8/9)
    r = .94

    Therefore Vmax = 1/3*pi*(8/9)*(4/3) = 1.241 cm^3


    Have i gone wrong somwhere
    Your calculation look OK. Except I'd give r = \frac{2 \sqrt{2}}{3} units and the maximum volume as \frac{32 \pi}{81} cubic units (you used cm^3 but the radius of the sphere is not given a unit in the question).
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  3. #3
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    Quote Originally Posted by Gul View Post
    A cone is inscribed in a sphere of radius 1, find the dimensions of the cone, and the maximum volume of the cone?

    I get a right right triangle:


    (h-1)^2 + r^2 = 1^2
    h^2 - 2h +1 + r^2 = 1

    r^2 = -h^2 + 2h

    r^2 = 2h - h^2

    Volume is: 1/3*pi*r^2*h

    V = 1/3*pi*(2h - h^2 )*h
    V = 1/3*pi*(2h^2-h^3)
    V' = 1/3*pi(4h - 3h^2) = 0
    V' = 4h - 3h^2 = 0 , h(4-3h) = 0 , h = 4/3

    r^2 = 2h - h^2

    r^2 = 2(4/3) - (4/3)^2

    r^2 = 8/3 - 16/9
    r^2 = 8/9
    r = sqrt(8/9)
    r = .94

    Therefore Vmax = 1/3*pi*(8/9)*(4/3) = 1.241 cm^3


    Have i gone wrong somwhere
    Of related interest: http://www.mathhelpforum.com/math-he...stion-3-a.html
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