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Math Help - Optimization Problem5

  1. #1
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    Optimization Problem5

    A cone-shaped drinking cup is made from a cirular piece of paper of radius R by cutting out a sector and joining the edge CA and CB. Find the maximum capacity of such a cup.

    The answer is attached.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by camherokid View Post
    A cone-shaped drinking cup is made from a cirular piece of paper of radius R by cutting out a sector and joining the edge CA and CB. Find the maximum capacity of such a cup.

    The answer is attached.
    Let the height of the cone be h and of the sector it is formed from be R.
    Then the radius of the base of the cone is

    r = sqrt(R^2-h^2)

    and so the volume of the cone is:

    V = (1/3) h pi r^2 = (pi/3) h (R^2-h^2)

    This volume is a maximum when dV/dh = 0, but:

    dV/dh = (pi/3) (R^2 - 3 h^2),

    which is zero when h = R/3, which is clearly a maximum, so the maximum
    volume V = (8 pi/27) R^3.

    RonL
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