How to Get the Maximum Volume from an Open Box that is to be made from a Rectangular sheet of any type of Material

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Given a Rectangular Sheet of any type of Material; same size Squares need to be cut out from the corners of the Rectangular Sheet, so as to make an Open Box. The Length of one side of the Square will be the Height of the Open Box. The Volume of the Open Box will vary depending on the length of the side of the Square. This Article will show How to Find the Ideal Height so as to obtain the Maximum Volume, and also why Calculus is needed in helping to solve this Problem.

Things You'll Need

  • Paper
  • Pencil
  • Pair of Sissors
  • 8 inches x 11 inches sheet of Paper
  • Tape and a
  • Calculator
  • The Material of choice, for this Example, will be a standard Rectangular Sheet of paper that can be found almost any where. The length is 11 inches and the width is 8 1/2 inches. We will take a pair of sissors and cut off 1/2 inch of the width, so that the Sheet of paper will have length of 11 inches and width of 8 inches ( the measurements are not to scale in the figure shown in the Image ). Please Click on the Image to see the figure.

  • We are going to see how the Volume of the Open Box changes as we cut out a square from each corner of the paper; the four squares will be of the same size. That is, in this Step #2, we will cut out four Squares; each Square having length of 1 inch. ( Please note that a Square is a four sided figure, in which the four sides are of the same length, and each side is perpendicular to each other.)Please Click on the Image to see the figure.

  • Now we bend the paper up, where the four dotted/broken lines are drawn, so as to form an Open Box. The Height of the Box will be 1 inch, the Width of the Box will be 6 inches, ( since 1 inch was taken away from each corner to make up the height ), and ( 8-2 ) = 6 inches, similarly, the length will be 9 inches ( 11-2 ) = 9 inches. The Formula to find the Volume of a Box is,.... The Product of ( The Height )x( The Length )x( The Width ). That is,... V = ( H )( L )( W ). So The Volume of this Open Box whose Height = 1 inch, Length = 9 inches, and Width = 6 inches is V = ( 1 )( 9 )( 6 ) = 54 cubic inches. Please Click on the Image to see the figure.

  • In the following Steps we will cut out Squares of lengths 2 inches and 3 inches, then we will make our Open Boxes and find their respective Volumes.

  • In this Step #5, we will cut out one Square from each corner of the paper; each Square having the length of 2 inches. Bend the paper up, where the four dotted/broken lines are drawn; so as to form an Open Box. The Height of the Box will be 2 inches, the Width of the Box will be 4 inches, and the lenghth of the box will be 7 inches( since 2 inches was taken away from each corner to make up the height. )
    The Width is ( 8-4 ) = 4 inches, and the Length is( 11-4 ) = 7 inches.
    So The Volume of this Open Box whose Height = 2 inches, length = 7 inches, and Width = 4 inches is V = ( 2 )( 7 )( 4 ) = 56 cubic inches. Please Click on the Image to see the figure.

  • In this Step #6, we will cut out one Square from each corner of the paper; each Square having length of 3 inches. Bend the paper up, where the four dotted/broken lines are drawn; so as to form an Open Box. The Height of the Box will be 3 inches, the Width of the Box will be 2 inches, and the length will be 5 inches ( since 3 inches was taken away from each corner to make up the height. ) The Width is( 8-6 ) = 2 inches, the length is ( 11-6 ) = 5 inches. So The Volume of this Open Box whose Height = 3 inches, Length = 5 inches, and Width = 2 inches is V = ( 3 )( 5 )( 2 ) = 30 cubic inches. Please Click on the Image to see the figure.

  • From the Steps above, we can see that the Volume of the Open Box changes as the Height changes. We now look for the Ideal/correct Height that will give us the Maximum Volume. To do so, In this Step #7, we will cut out One Square from each corner, each Square having length of x inches, where x has to be greater than Zero ( 0 ) and less than Four ( 4 ) inches, so that a three-dimensional box can be made. We then bend the paper up, where the four dotted/broken lines are drawn, so as to form an Open Box. The Height of the Box will be x inches, the Width of the Box will be (8 - 2x) inches and the Length will be (11 - 2x)inches ( since 2, x inches was taken away from each corner to make up the height.)
    So The Volume of this Open Box whose Height = x inches, width = (8-2x) inches, and Length = (11-2x) inches is; V(x) = ( x )( 11-2x )( 8-2x ) cubic inches,
    where V(x) is, the Volume expressed in terms of the variable x. ( Read as ' The function of V of x ', and NOT ' The Product of V multiplied by x '. ) . Please Click on the Image to see the figure.

  • So V(x) = ( x )( 11-2x )( 8-2x )
    V(x) =( 11x - 2x^2 )( 8 - 2x )
    V(x) = 88x - 22x^2 - 16x^2 + 4x^3
    V(x) = ( 88x - 38x^2 + 4x^3 ) cubic inches.
    In order to find the Exact Height x that will give us the Maximum Volume we have to use Calculus, so as to Minimize Time. Otherwise, we can choose all the values for x that are between Zero ( 0 ) and ( 4 ), and substitute each value into the Function V(x) and get the respective values for the Volume. We are not including ( 0 ) and ( 4 ) since these two values will give us no Open BOX. that is, ( 0 ), will give us the original flat Sheet of paper, and ( 4 ), will give us a folded Sheet of Paper with Height = 4 inches and Length = 3 inches, and NO Width (that is Width = 0 inches.)

  • We now take the Derivative of V(x), written V'(x), set it equal to Zero ( 0 ), and Solve for x. The value of x that is between ( 0 ) and ( 4 ), we will substitute into the Function V(x), and get the Maximum Volume. In the following Step #10, We find V'(x).

  • V(x) = 4x^3 - 38x^2 + 88x, [Note: If V(x) = Ax^n, then V'(x)=Anx^(n-1)].
    V'(x) = 12x^2 - 76x + 88,
    Let V'(x) = 0, then
    0 = 12x^2 - 76x + 88
    0 = 4( 3x^2 - 19x + 22 )
    0 = 3x^2 - 19x + 22
    By the Quadractic Formula, x = ( -b +/-square root(b^2-4ac))/(2a)
    x = ( 19 + Sq.Rt( 97 ))/6 = 4.81 or x = ( 19 - Sq.Rt.( 97 ))/6 = 1.525
    Since ( 1.525 ) is between ( 0 ) and ( 4 ) then x = 1.525 is the Height of the Open Box that will get the Maximum Volume.

  • The Volume of the Open Box with
    Height = 1.525, Length = ( 11 - (2)(1.525)) and Width = ( 8-(2)(1.525))
    is,... V= (1.525)(7.95)(4.95) = 60.012 Cubic Inches.

Tips & Warnings

  • A general formula to find the Maximum Volume of an Open Box, that is made from any material of a Rectangular Shape, given the Length, L, and the Width, W,
  • is to find the correct Height, H. The Formula, for the correct H is,...
  • H ={[L+W]+Sq.Rt.[(L+W)^2 - 3LW]}/6, where H is greater than Zero ( 0 ) and less than one-half the Width, W, if the Width is less than the Length, or one-half of the Length, L, if the Length is less than the Width.
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