Question 388388
The piece of plastic is 15" by 10".
The squares cut from each corner are x" by x".
The volume of the box made from folding up the sides after the square corners have been removed can be expressed by {{{V = x(B)}}} where B is the area of the bottom of the box and x is the height of the box sides:
The area of the box bottom can be expressed by:
{{{B = (10-2x)(15-2x)}}}, so the volume equation becomes:
{{{V = x(10-2x)(15-2x)}}}
{{{V = x(150-50x+4x^2)}}} or...
{{{V = 4x^3-50x^2+150x}}} This is the equation for the volume of the box.
The graph of this cubic equation is:
{{{graph(400,400,-5,10,-50,140,4x^3-50x^2+150x)}}}
From the graph, you can see that the relative maximum (of y, the volume) occurs at about x = 2 (it's really 1.96187390745...), so the dimensions of the square cut-outs would be 2" by 2" to obtain the maximum volume.