posted by jim .
You are planning to make an open-top box from an 12 in by 12 in piece of cardboard
by cutting congruent squares from the corners and folding up the sides.
What are the dimensions (of the 3 sides) of the largest volume you can make this way?
let the size of the cut out squares be x by x inches
so the base of the box is 12-2x by 12-2x and its height is x inches.
volume = x(12-2x^2
expand, then find d(volume)dx
set that equal to zero and solve for x,
from which you can then find the dimensions