You cut square corners with side lengths that are whole numbers from a piece of cardboard with dimensions 20 inches by 30 inches. You then fold the cardboard to create a box with no lid. Which of the following dimensions will give you the greatest volume?

Suppose you cut squares have the side lengths x then we have

length of base box=20-2x

width of base box=30-2x

height of box=x

volume of the box

V=x (20-2x) (30-2x)

V=600x-100x^2+4x^3

then

dV/dx=600-200x+12x^2

set that to zero

600-200x+12x^2=0

apply quadratic formula

x=approx12.97 or 3.92

since you want in whole numbers so x=13 or x=4

so if x=13 then V = - 312 but negative number doesn't make a sense so we ignore that

if x=4 then v=1056 this is the greatest volume

the dimensions are

length of a box base=12

width of a box base=22

height of box = 4

did u get it?