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16 November, 07:02

A cylindrical package with half a hemisphere to be sent by a postal service can have a maximum combined length (height of the cylinder) and girth (perimeter of a cross section of the cylinder) of 108 inches. Find the dimensions of the package of maximum volume that can be sent. Assume the cross section is a circle. (Make sure you verify that the value you found is the maximum).

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Answers (2)
  1. 16 November, 09:30
    0
    36 x 18 x 18

    Step-by-step explanation:

    By Lagrange multiplier's method,

    ∇f (x, y, z) = λ∇g (x, y, z) & g (x, y, z) = k

    considering x, y, z as three unequal sides of the box

    f (x, y, z) = xyz

    If x represents the length, then constraint condition is g (x, y, z) = x + 2 (y+z) = 108

    We have the following three equations:

    yz = λ - ->eq (1) fx = λgx

    xz=2λ - ->eq (2) fy=λgy

    xy = 2λ - ->eq (3) fz=λgz

    x+2y+2z=108-->eq (4)

    Dividing eq (2) by eq (1), we'll have

    x / y = 2

    => x = 2y

    Also, by using eq (2) and eq (3), we can represent y=z

    By substituting 'x = 2y' and 'y=z' in eq (4), we have

    eq (4) = >

    2y + 2 (y+y) = 108

    y=108/6

    y=18

    For x: x=2y=> 2 (18) = >36

    For z: z = y = 18

    Therefore, the dimensions are 36 x 18 x 18
  2. 16 November, 10:59
    0
    Radius of package is 36/π inches

    Height of package is 36 inches

    Step-by-step explanation:

    Here we have

    Sum of height plus perimeter of package = 108 inches that is

    2πr + h = 108 inches

    Where:

    Height of package = h

    Radius of package = r

    Volume of package = πr²h

    However, h = 108 - 2πr

    Therefore, the equation for the volume of the package is;

    Volume of package, V = πr² (108 - 2πr) = πr²*108 - 2π²r³

    Differentiating the above equation and equating to zero to find a maximum value, we have;

    2πr*108 - 6r²π² = 0

    2πr*108 = 6r²π²

    36 = r·π

    r = 36/π inches

    h = 108 - 2πr = 108 - 72 = 36 inches

    Radius of package, r = 36/π inches

    Height of package, h = 36 inches.
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