you are given a piece of cardboard 14 inches long and 8 inches wide. you want to create an open topped box by cutting x-inch squares out of the corners and folding up the sides so the edges meet. what are the approximate dimensions of the box with the maximum volume(to the nearest quarter-inch)? what is the maximum volume of the box (to the nearest tenth of a cubic inch)?

The space occupied inside an object's borders in three dimensions is referred to as its volume. It is sometimes referred to as the object's capacity. An object's volume is the amount of space it occupies. Volume, which is measured in cubic units, is the 3-dimensional space occupied by matter or encircled by a surface. The quantity of space that anything occupies is measured by its volume. Volume measures include things like cups of flour, gallons of milk, and cubic feet of helium. The heaviness of an object is gauged by its weight.

Here,

V=lbh

14-2x=l

8-2x=b

x=h

V=x(14-2x)(8-2x)

V=x(112-16x-28x+4x²)

V=4x³-44x²+112x

V'=12x²-88x+112

on solving the quadratic equation,

x=5.69,1.64

V''=24x-88

Since x<2, so we will choose 1.64.

24*1.64-88<0

-48.64<0

h=1.64 inch

l=14-2x

=14-2*1.64

=10.72 inch

b=8-2x

=8-2*1.64

=4.72 inch

Maximum volume=lbh

=1.64*4.72*10.72

=82.98 inch³

The dimension of box will be length is 10.72 inch, width is 4.72 inch and height is 1.64 inch. The volume of the box is 82.98 inch³.

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