An open rectangular box is to be made from a piece of cardbo

An open rectangular box is to be made from a piece of cardboard 8 inches wide and 8 inches long by cutting a square from each corner and bending up the sides.

1. express the volume of the box as a function of the size x of the cutout

2. approximate the dimensions of the box with the largest volume

Solution

Length of the box = 8-2x
Width of the box = 8-2x
Height of the box = x

Volume of the box:
V = (8-2x)(8-2x)x
V= 64x - 32x² + 4x³

2).

to get the largest volume

dV/dx=0

dV/dx= 64 -64x+12x^2

12x^2 -64x +64 =0

4(3x^2 -16x +16)=0

3x^2 -16x +16 =0

3x^2 -12x -4x +16=0

3x(x-4) -4(x-4) =0

so (3x-4) (x-4) =0

so x = 4(reject) or x =3/4

x=4 is not for large volume

the dimensions are 8 - 2x3/4 , 8-2x3/4 , 3/4

                                13/2 ,13/2, 3/4