You are constructing a cardboard box from a piece of cardboard with the dimensions 2 m by 4 m. You then cut equal-size squares from each cor

Question

You are constructing a cardboard box from a piece of cardboard with the dimensions 2 m by 4 m. You then cut equal-size squares from each corner so you may fold the edges. What are the dimensions (in m) of the box with the largest volume

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Maya 2 weeks 2021-09-13T22:49:07+00:00 1 Answer 0

Answers ( )

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    2021-09-13T22:50:17+00:00

    Answer:

    Dimensions:

    l =3,15 m

    w=1,15 m

    x= 0,42 m   (height)

    V(max) = 1,52 m³

    Step-by-step explanation:

    The cardboard is L = 4 m   and   W = 2 m

    Let call x the length of the square to cut in each corner, then, volume of open box is:

    For the side L       is  L – 2*x               l =  4 – 2*x

    For the side W      is  W – 2*x             w=  2 – 2*x

    The height              is  x

    Volume of the open box, as function of x is:

    V(x) = ( 4 -2x) * ( 2 – 2x) *x    ⇒  V(x) = ( 8 – 8x -4x + 4x²) *x

    V(x) = ( 8 -12x + 4x² )*x     V(x) = 8x – 12x² + 4x³

    V(x) = 8x – 12x² + 4x³

    Taking derivatives on both sides of the equation

    V´(x) = 8 – 24x + 12x²

    V´(x) = 0       8  – 24x + 12x² = 0     reordering    12x² – 24x + 8  = 0

    or    3x² – 6x + 2 = 0

    A second degree equation. Solving for x

    x₁,₂  = ( 6 ± √36 – 24 ) /6

    x₁,₂  = ( 6 ± 3.46) /6

    x₁  =  6 + 3,46 /6     x₁  = 1.58  we dismiss such solution because 1,58 * 2 = 3,15 and is bigger than 2 one of the side of the cardboard

    x₂  =( 6 – 3,46 ) / 6

    x₂  = 0,42 m

    Dimensions of the open box

    l  = 4 – 2*x     l  = 4 – 0,85      l  =  3,15 m

    w = 2 -2*x    w  = 2 – 0,85     w = 1,15 m

    x = 0,42 m

    V(max) =3,15*1,15*0,42

    V(max) = 1,52 m³

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