A box is to be made out of a 12 by 20 piece of cardboard. Squares of equal size will be cut out of each corner, and then the ends and sides

Question

A box is to be made out of a 12 by 20 piece of cardboard. Squares of equal size will be cut out of each corner, and then the ends and sides will be folded up to form a box with an open top. Find the length L , width W , and height H of the resulting box that maximizes the volume.

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Thành Đạt 7 months 2021-08-18T11:24:29+00:00 1 Answers 3 views 0

Answers ( )

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    2021-08-18T11:25:35+00:00

    Answer:

    Box Dimensions:

    L  = 15.15 ul

    W = 7.15 ul

    h = x = 2.43 ul

    V(max) =    263.22 cu

    Step-by-step explanation:

    We call x the length of the square to be cut in the corners then:

    Are of the base of the box is:

    (20 – 2*x)  is the future length of the box and

    (12 – 2*x) will be the width

    The heigh is x  then the volume of the box is:

    V = ( 20 – 2*x )* ( 12 – 2*x ) * h

    And the volume as a function of x is:

    V(x) = ( 20 – 2*x) * ( 12 – 2*x ) * x     or   V(x) = (240 -40*x -24*x + 4*x²) * x

    V(x) =  240*x – 64*x² + 4*x³

    Taking derivatives on both sides of the equation we get:

    V´(x) = 240 – 128*x + 12*x²

    V´(x)  =  0              240 – 128*x + 12*x²  = 0    or     60 – 32*x + 3*x²

    3*x²  –  32*x  +  60  = 0

    Solving:

    x₁,₂  =  32  ± √ (32)² – 4*3*60 ]/ 2*3

    x₁,₂  =  32  ± √ 1024 – 720 )/6

    x₁,₂  = ( 32  ± √ 304 )/6

    x₁,₂  = ( 32  ± 17.44 )/6

    x₁  =  8.23      ( we dismiss this solution because is not feasible 2*x > 12

    x₂ = 2.43  u.l ( units of length)

    Then

    L  =  20 – 2*x      L  = 20  – 4.85     L  = 15.15 ul

    W =  12 – 2*x      W = 12 – 4.85      W = 7.15 ul

    h  =  2.43 ul

    V = 2.43*7.15*15.15  cubic units

    V =  263.22 cu

    To see if when x = 2.43   function V has a maximum we go to the second derivative

    V´´(x)  = – 128  + (24)*2.43

    V´´(x) = – 69.68      as        V´´(x)  < 0   then we have a maximum for V(x) in the point x = 2.43

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