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An open box is to be made out of a 8-inch by 16-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. Find the dimensions of the resulting box that has the largest volume.
Dimensions of the bottom of the box: (?) x (?)
Height of the box: (?)

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Answer:

Dimension of bottom =Length x Breadth

                                   =  12.619 inch  x 4.619‬ inch

                                   ≈ 13 inch x 5 inch  (To nearest whole number)

Height of box = 1.6905 inch

                       ≈ 2 inch

Step-by-step explanation:

Length of the cardboard = 16 inch

Breadth of the cardboard = 8 inch

Since the cardboard has four corners, we would cut out four (4 ) squares.

Assuming the  length of the square =y

length of open box    = 16-2y----------------------------------- (1)

breadth of open box  = 8-2y-------------------------------------(2)

height of open box    = y

Volume of box = Length x breadth x height

        =   (16-2y) x ( 8-2y) x y

        =   ( 128-48y + 4y²)y

        =   128y-48y² + 4y³

To maximize the volume of the box,  V(y); dV/dy =0

dV/dy = 128- 96y + 12y²

12y²-96y+128 = 0

a = 12 , b= -96 , c= 128

y= [-b±  √(b²-4ac)]/2a

=  [96 ±√(96²- (4 x 12 x128)]/2 (12)

y= 6.3094 or y =1.6905

Substituting both values of y into (1) and (2)

When y =  6.3094

Length of box = 16- 2(6.3094)

                                  = 3.3812 inch

Breadth of box =8- 2 (6.3094)

                         = -4.6188‬ inch

Since the breadth cannot be negative, this value of y is therefore an extraneous root.

So the correct value of y is the smaller of the two roots of equation,

y = 1.6905 inch

Substituting both values of y into (1) and (2)

When y =  1.6905

Length of box = 16- 2(1.6905)

                                  = 12.619 inch

                                  ≈ 13 inch

Breadth of box =8- 2 (1.6905)

                         = 4.619‬ inch

                         ≈ 5 inch

So the Dimension of the  bottom (to nearest whole number):

Length x breadth =  13 inch x  5 inch

Height of the box =  1.6905 inch

                             ≈  2 inch

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