Question

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?

Answer 1

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 ?